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UNIVERSITY  OF  CALIFORNIA 
AT   LOS  ANGELES 


A  TREATISE 


ADJUSTMENT  OF  OBSERVATIONS, 


APPLICATIONS  TO  GEODETIC  WORK  AND  OTHER  MEASURES 
OF  PRECISION. 


T.  W.  WRIGHT,  B.A., 

Civil  Engineer, 

LATE    ASSISTANT   ENGINEER    UNITED    STATES    LAKE    SURVEY. 


O  Messkunst  Zaum  der  Phantasie, 
Wtr  dir  will  folgeit  irrct  nie. 


— HALI.ER. 


NEW  YORK : 

D.  VAN  NOSTRAND, 

23  MURRAY  STREET  AND  27  WARREN  STREET. 

1884. 


Copyright, 

1884. 
T.  W.  WRIGHT. 


PRINTED  BY 

H.  J.  HEWITT, 

27  ROSE  STREET,  NEW  YORK. 


Q. 


PREFACE. 


IN  the  following  treatise  I  have  endeavored  to  give  a 
systematic  account  of  the  method  of  adjusting  observations 
founded  on  the  principle  of  the  mean.  The  more  important 
applications,  especially  with  reference  to  geodetic  and 
astronomical  work,  are  fully  discussed. 

It  has  been  my  aim  throughout  to  be  practical.  The 
book  originated  and  grew  amid  actual  work,  and  hence 
subjects  that  are  interesting  mainly  because  they  are  curious, 
and  methods  of  reduction  that  have  become  antiquated,  are 
not  noticed. 

Several  of  the  views  enunciated  are  not  in  the  usual 
strain,  but  they  are,  however,  such  as  I  think  all  experienced 
observers,  though  perhaps  not  all  mathematicians,  will  at 
once  assent  to. 

As  regards  notation  I  have  been  conservative,  usually 
following  Encke's  system  as  given  by  Chauvenet.  Some 
minor  changes  have  been  introduced  which  it  is  thought 
will  tend  to  greater  clearness  and  uniformity  of  expression. 

The  examples  and  illustrations  have  been  drawn  chiefly 
from  American  sources,  for  the  reason  that  much  valuable 
material  of  this  kind  is  to  be  had,  and  that  thus  far  it  has 
not  been  used  for  this  purpose.  They  have  been  taken 
from  records  of  work  actually  done,  and  principally  irom 
work  with  which  I  have  been  connected. 

2 

331       211005 


4  PREFACE. 

In  the  applications  to  practical  work  1  have  aimed  at 
giving  only  so  much  of  methods  of  observing  as  would 
serve  to  make  the  methods  of  adjustment  intelligible.  It 
has  been  difficult  to  do  this  succinctly  and  at  the  same  time 
satisfactorily,  and  accordingly  references  are  given  to  books 
where  descriptions  of  instruments  and  modes  of  using  them 
can  be  found. 

Special  attention  has  been  given  to  the  explanation  of 
checks  of  computation,  of  approximate  methods  of  adjust- 
ment, and  of  approximate  methods  of  finding  the  precision 
of  the  adjusted  values.  But  in  order  to  see  how  far  it  is 
allowable  to  use  these  short  cuts  the  rigid  methods  must 
first  be  derived.  It  is  for  this  reason  principally  that  the 
subject  of  triangulation  has  been  dwelt  on  at  such  length. 
In  general  it  is  unnecessary  to  spend  a  great  amount  of 
time  in  finding  the  probable  error,  when,  after  it  has  been 
found,  it  in  many  cases  tells  so  little. 

I  have  been  careful  to  give  references  to  original  author- 
ities as  far  as  I  could  ascertain  them,  and  also  to  give  lists 
of  memoirs  on  special  subjects  which  will  be  of  use  to  any 
one  desiring  to  follow  those  subjects  farther.  Of  recent 
writers  I  am  indebted  chiefly  to  Helmert  and  Zacharise.  I 
desire  also  to  acknowledge  my  obligations  to  my  old  Lake 
Survey  friends,  Messrs.  C.  C.  Brown,  J.  H.  Darling,  E.  S. 
Wheeler,  R.  S.  Woodward,  and  A.  Ziwet,  who  have  read 
the  manuscript  in  part  and  given  me  the  benefit,  of  their 
advice.  Mr.  Brown  deserves  special  mention  for  assistance 
rendered  while  the  book  was  passing  through  the  press. 


TABLE    OF  CONTENTS. 


CHAPTER  I. 

Introduction. 

General  remarks  on  observing: 

The  instrument,          .......... 

External  conditions,       ......... 

The  observer,      .         .         . 

Synopsis  of  mathematical  principles  employed  : 

Theory  of  probability,    .........  18 

Definite  integrals.       ..........         19 

Taylor's  theorem — Examples 21 

Interpolation,      ...........         25 

Periodic  series,       ..........  27 

Notation,    ............         28 


CHAPTER  II. 

T^i?  Law  of  Error. 

The  arithmetic  mean  : 

Quantity  measured  the  quantity  to  be  found,         ....  29 

The  arithmetic  mean  the  most  plausible  value.  ...  31 

When  the  arithmetic  mean  gives  the  true  va'ue,     ...  32 

Inferences  from  the  arithmetic  mean,          .....  33 

Quantity  measured  a  function  of  the  quantities  to  be  found,         .  34 

The  most  plausible  values  of  the  unknowns,      ....  37 

Law  of  error  of  a  single  observed  quantity,  .....  38 

The  principle  of  least  squares,  .......  42 

Reduction  of  observations  to  a  common  basis,      ....  44 

Combination  of  heterogeneous  measures,  .....  44 

Law  of  error  of  a  linear  function  of  independently  observed  quantities,  44 


6  THE   ADJUSTMENT   OF   OBSERVATIONS. 

PAGE 

Comparison  of  the  accuracy  of  different  series  of  observations: 

The  mean-square  error,         ........  46 

The  probable  error,  ..........  47 

The  average  error,         .........  48 

The  probability  curve,     ..........  54 

The  law  of  erior  applied  to  an  actual  series  of  observations: 

Effect  of  extending  the  limiis  of  error  to  ±  OD  ,      ....  57 

Various  laws  of  error,       .........  59 

Experimental  proof  of  the  exponential  law,         ....  65 

General  conclusions,         .........  66 

Classification  of  observations,     ........  67 

CHAPTER  III. 

Adjtistinent  of  Direct  Observations  of  One   Unknown. 

Observed  values  of  equal  quality  : 

The  most  probable  value — The  arithmetic  mean,     ....  69 

Control  of  the  arithmetic  mean,     ......  71 

Precision  of  the  arithmetic  mean  : 

Bessel's  formula,          ........  73 

Peters'  formula,       ........  76 

Approximate  formulas,       .......  79 

The  law  of  error  tested  by  experience,         .....  82 

Caution  as  to  the  tests  of  precision,          ......  85 

Constant  error,      ..........  89 

Necessary  closeness  of  computation,        ......  gi 

Observed  values  of  different  quality  : 

The  most  probable  value — The  weighted  mean,           ••       •         •  92 

Combining  weights,            ........  94 

Reduction  of  observations  to  a  common  standard,       .         .  95 

Computation  of  weights,   .........  96 

Control  of  the  weighted  mean,       ......  96 

Precision  of  the  weighted  mean — Control  of  [/z^'J,          .         .  99 
Observed  values  multiples  of  the  unknown  : 

The  most  probnble  value,     ........  102 

Precision  of  a  linear  function  of  independently  observed  values,           .'  105 

Miscellaneous  examples,     .........  no 

NOTE  I. — On  the  weighting  of  observations,            .....  118 

An  approximate  method,      ........  120 

Weighting  when  constant  error  is  present, 121 

Assignment  of  weight  arbitrarily,         ......  126 

Combination  of  good  and  inferior  work,          .....  127 

The  weight  a  function  of  our  knowledge,     .....  128 

General  remarks,       ..........  130 

NOTE  II. — On  the  rejection  of  observations,       .....  131 


TABLE   OF   CONTENTS.  7 

CHAPTER  IV. 

Adjustment  of  Indirect  Oi>ser;>atioiis. 

PAGE 

Determination  of  the  most  probable  values,   ......  139 

Formation  of  the  normal  equations,     ......  145 

Control  of  the  formation,  ........  150 

Forms  of  computing  the  normal  equations: 

With  multiplication  tables  or  a  machine,        .         .         .  151 

With  a  table  of  logarithms, 152 

With  a  table  of  squares,           ......  154 

Solution  of  the  normal  equations: 

The  method  of  subsiitution,       .......  156 

Controls  of  the  solution, 158 

Forms  of  solution  : 

Solution  without  logarithms,        .....  160 

Solution  with  logarithms,          .....  163 

The  method  of  indirect  elimination,           .....  165 

Combination  of  the  direct  and  indirect  methods,          .         .  167 

Time  required  to  solve  a  ?et  of  equations,         ....  172 

Precision  of  the  most  probable  (adjusted)  values,     ....  174 

First  method  of  finding  the  weights,         ......  177 

Special  case  of  2  and  3  unknowns,       .....  178 

Modifications  of  the  general  method,        .....  180 

Second  method  of  finding  the  weights,        .....  184 

To  find  the  m.  s.  e.  of  a  single  observation,     .....  186 

Methods  of  computing  [?'?']>           ......  187 

Precision  of  any  function  of  the  adjusted  values  (three  methods),         .  193 
Average  value  of  the  ratio  of  the  weight  of  an  observed  value  to  its 

adjusted  value,           ..........  198 

Examples  and  artifices  of  elimination,       ......  201 


CHAPTER  V. 

Adjustment  of  Condition  Observations. 

General  statement,           ..........  213 

Direct  solution — Method  of  independent  unknowns,         .         .         .  214 

Indirect  solution — Method  of  correlates,        ......  217 

Precision  of  the  adjusted  values  or  of  any  function  of  them,      .  224 

Mean-square  error  of  an  observation  of  weight  unity,      .         .  224 

Weight  of  the  function,           .......  227 

Solution  in  two  groups,            .........  238 

Programme  of  solution,        ........  242 

Precision  of  the  adjusted  values  or  of  any  function  of  them,            .  243 

Solution  by  successive  approximaiion,       ......  247 


8  THE   ADJUSTMENT   OF   OBSERVATIONS. 

CHAPTER  VI. 

Application  to  the  Adjustment  of  a   Triangulation. 

PAGE 

General  statement,           . 250 

The  method  of  independent  angles,    .......  252 

The  local  adjustment,       .........  255 

Number  of  local  equations,             .         .                   .         .         .  258 

The  general  adjustment,            ........  259 

The  angle  equations,      .         .         .         .         .         .         .         .  259 

Number  of  angle  equations,        ......  261 

The  side  equation's,        ........  263 

Reduction  to  the  linear  form,     ......  266 

Check  computation,        .......  268 

Position  of  pole,          ........  271 

Number  of  side  equations,      ......  272 

Check  of  the  total  number  of  conditions,          ....  273 

Manner  of  selecting  the  angle  and  side  equations,       .         .  273 
Adjustment  of  a  quadrilateral : 

Solution  by  independent- unknowns,         .         .         .         .         .  280 

Precision  of  the  adjusted  values,  .....  282 

Solution  by  correlates,        ........  284 

Precision  of  the  adjusted  values,    .....  286 

Solution  in  two  groups,     ........  288 

Precision  of  the  adjusted  values,    .....  293 

Solution  by  successive  approximation,     .....  298 

Adjustment  of  a  triangulation  net,       ......  300 

Artifices  of  solution,           ........  302 

The  local  adjustment,      .......  302 

The  general  adjustment,     .......  303 

Approximate  method  of  finding  the  precision,     ....  3*3 

The  method  of  directions,        .........  315 

The  local  adjustment,           .         .         .         .         .         .         .         .  316 

Checks  of  the  normal  equations, 32° 

Precision  of  the  adjusted  values,  ......  322 

The  general  adjustment,           ........  323 

Approximate  method  of  reduction,       ......  32^ 

Modified  rigorous  solution  : 

General  statement,    ..........  33r 

Local  adjustment,         .         .         .      • 333.336 

General  adjustment, 336>  339 

On  breaking  a  net  into  sections,          .......  341 

Adjustment  for  closure  of  circuit,   .         .         .         .         .         •         •         •  342 

Discrepancy  in  azimuth,       . 343 

Disci  epancy  in  bases, 347 

Discrepancy  in  latitude  and  longitude,        .....  347 


TABLE   OF   CONTENTS.  9 

CHAPTER  VII. 

Application  to  Base-Line  Measurements. 

PAGE 

General  statement 349 

Precision  of  a  base-line  measurement,        ......  350 

The  Bonn  Base, 353 

The  Chicago  Base, 356 

Length  and  number  of  bases  necessary  in  a  triangulation, .  .         .         .  356 

Conneciion  of  a  base  with  the  main  trinngulation,     ....  360 

Adjustment  of  a  triangulation  when  more  than  one  base  is  considered,  362 

Rigorous  solution,    ..........  363 

Approximate  solutions  : 

When  the  angles  alone  are  changed,     .....  365 

When  the  bases  alone  are  changed,           .....  368 

CHAPTER  VIII. 

Appliiation  to  Levelling. 

Spirit  levelling,    ...........  371 

Precision  of  a  line  of  levels,     ........  374 

Adjustment  of  a  net  of  levels,  .......  377 

Approximate  methods  of  adjustment,       .....  378 

Trigonometrical  levelling,           ........  382 

To  find  the  refraction  factors,            .......  384 

To  find  the  mean  coefficient  of  refraction,    .....  386 

To  find  the  differences  of  height 386 

Precision  of  the  differences  of  height,  ....  388 

Adjustment  of  a  net  of  trigonometrical  levels,  ....  388 

Approximate  methods  of  adjustment,  .....  392 

CHAPTER  IX. 

Application    to   Errors  of  Graduation  of  Line   Measures  and  to  Calibration   of 

Thermometers. 

Line  measures,         ...........  395 

Calibration  of  thermometers,       ........  401 

CHAPTER  X. 

Application  to  Empirical  Formulas  and  Interpolation. 

General  statement,           .         .         .         .         .         .         .         .         .         .  408 

Applications,          ..........  413 

Periodic  phenomena,       ..........  417 

Applications, 421 

APPENDIX  I.— Historical  note,         ....  427 

APPENDIX  II. — The  law  of  error,         ....  429 

TABLES,                    435 


THE  ADJUSTMENT  OF  OBSERVATIONS, 


CHAPTER  I. 

INTRODUCTION. 

THE  factors  that  enter  into  the  measurement  of  a  quan- 
tity are,  the  observer,  the  instrument  employed,  and  the  con- 
ditions under  which  the  measurement  is  made. 

i.  The  Instrument. —  If  the  measure  of  a  quantity  is 
determined  by  untrained  estimation  only,  the  result  is  of 
little  value.  The  many  external  influences  at  work  hinder 
the  judgment  from  deciding  correctly.  For  example,  if  we 
compare  the  descriptions  of  the  path  of  a  meteor  as  given 
by  a  number  of  people  who  saw  the  meteor  and  who  try  to 
tell  what  they  saw,  it  would  be  found  impossible  to  locate 
the  path  satisfactorily.  The  work  of  the  earlier  astrono- 
mers was  of  this  vague  kind.  There  was  no  way  of  testing 
assertions,  and  theories  were  consequently  plentiful. 

The  first  great  advance  in  the  science  of  observation  was 
in  the  introduction  of  instruments  to  aid  the  senses.  The 
instrument  confined  the  attention  of  the  observer  to  the 
point  at  issue  and  helped  the  judgment  in  arriving  at  con- 
clusions. As  with  a  rude  instrument  different  observers 
would  get  the  same  result,  it  is  not  to  be  wondered  at  that 
for  a  long  time  a  single  instrumental  determination  was 
considered  sufficient  to  give  the  value  of  the  quantity 
measured. 

The  next  advance  was  in  the  questioning  of  the  instru- 
ment and  in  showing  that  a  result  better  on  the  whole  than 
a  single  direct  measurement  could  be  found.  This  opened 

3 


12  THE  ADJUSTMENT  OF  OBSERVATIONS. 

the  way  for  better  instruments  and  better  methods  of  ob- 
servation. For  example,  Gascoigne's  introduction  of  cross- 
hairs into  the  focus  of  the  telescope  led  to  better  graduated 
circles  and  to  better  methods  of  reading  them,  resulting 
finally  in  the  reading  microscopes  now  almost  universally 
used.  The  culminating  point  was  reached  by  Bessel,  who, 
by  his  systematic  and  thorough  investigation  of  instrument- 
al corrections  and  methods  of  observation,  may  be  said  to 
have  almost  exhausted  the  subject.  He  confined  himself, 
it  is  true,  to  astronomical  and  geodetic  instruments,  but  his 
methods  are  of  universal  application. 

The  questioning  of  an  instrument  naturally  arises  from 
noticing  that  there  are  discrepancies  in  repeated  measure- 
ments of  a  magnitude  with  the  same  instrument  or  in  meas- 
ures made  with  different  instruments.  Thus,  if  a  distance 
was  measured  with  an  ordinary  chain,  and  then  measured 
with  a  standard  whose  length  had  been  very  carefully  de- 
termined, and  the  two  measurements  differed  widely,  we 
should  suspect  the  chain  to  be  in  error  and  proceed  to  ex- 
amine it  before  further  measuring.  So  discrepancies  found 
in  measurements  made  with  the  same  measure  at  different 
temperatures  have  shown  the  necessity  of  finding  the  length 
of  the  measure  at  some  fixed  temperature,  and  then  apply- 
ing a  correction  for  the  length  at  the  temperature  at  which 
the  measurement  is  made. 

Corrections  to  directly  measured  values  are  thus  seen  to 
be  necessary,  and  to  be  due  to  both  internal  and  external 
causes.  The  internal  causes  arising  from  the  construction 
of  the  instrument  are  seen  to  be  in  great  measure  capable 
of  elimination.  From  geometrical  considerations  the  ob- 
server can  tell  the  arrangement  of  parts  demanded  by  a 
perfect  instrument.  He  can  compute  the  errors  that  would 
be  introduced  by  certain  supposed  irregularities  in  form 
and  changes  of  condition.  The  instrument-maker  cannot, 
it  is  true,  fulfil  the  conditions  necessary  for  a  perfect  instru- 
ment, but  he  has  been  gradually  approaching  them  more 
and  more  closely.  It  is  to  be  remembered  that,  even  if  an 


INTRODUCTION.  13 

instrument  could  be  made  perfect  at  any  instant,  it  would 
not  remain  so  for  any  great  length  of  time. 

It  hence  followed  as  the  next  great  advance  that  the 
instrument  was  made  adjustable  in  most  of  its  parts,  so  that 
the  relative  positions  of  the  parts  are  under  the  control  of 
the  observer.  This  is  getting  to  be  more  and  more  the 
case  with  the  better  class  of  instruments. 

Not  only  is  error  diminished  by  the  improved  construc- 
tion of  the  instrument,  but  also  by  more  refined  methods  of 
handling  it.  It  may  be,  indeed,  that  some  contrivances 
beyond  those  required  to  make  necessary  readings  for  the 
measure  of  the  quantity  in  question  may  be  needed.  Thus, 
with  a  graduated  circle  regular  or  periodic  errors  of  gradu- 
ation may  be  expected.  If  the  angle  between  two  signals 
were  read  with  a  theodolite,  the  reading  on  each  signal,  and 
consequent  value  of  the  angle,  would  be  influenced  by  the 
periodic  errors  of  the  circle  of  the  instrument.  Though  a 
single  vernier  or  microscope  would  suffice  to  read  the  cir- 
cle when  the  telescope  is  directed  to  the  signals,  yet,  as  the 
circle  is  incapable  of  adjustment,  we  can  only  get  rid  of  the 
influence  of  the  periodicity  by  employing  a  number  of  ver- 
niers or  microscopes  placed  at  equal  intervals  around  the 
circle.  It  happens  that  this  same  addition  of  microscopes' 
eliminates  eccentricity  of  the  graduated  circle  as  well. 

This  same  principle  of  making  the  method  of  observation 
eliminate  the  instrumental  errors  is  carried  through  even 
after  the  nicest  adjustments  have  been  made.  Thus,  in  or- 
dinary levelling,  if  the  backsights  and  foresights  are  taken 
exactly  equal  the  instrumental  adjustment  may  be  poor  and 
still  good  work  may  be  done.  But  good  work  is  more 
likely  if  the  adjustments  have  been  carefully  made,  as  if  for 
unequal  sights,  and  still  the  sights  are  taken  equal. 

Simplicity  of  construction  in  an  instrument  is  also  to  be 
aimed  at.  An  instrument  that  theoretically  ouglit  to  work 
perfectly  is  often  a  great  disappointment  in  practice.  Two 
striking  examples  are  the  compensating  base-apparatus  and 
the  repeating  theodolite,  both  of  which  have  been  aban- 


14  THE  ADJUSTMENT   OF   OBSERVATIONS. 

doned  on  all  the  leading  surveys.  In  both  cases  the  me- 
chanical difficulties  in  the  way  have  proved  insurmount- 
able, and  the  instruments  have  been  replaced  by  others  of 
simpler  construction,  to  whose  readings  corrections  can 
either  be  computed  and  applied  or  the  errors  of  the  read- 
ings can  be  eliminated  by  the  me'thod  of  observation.  In 
this  way  no  hopes  of  an  accuracy  which  cannot  be  realized 
are  held  out. 

Such  is  the  perfection  now  attained  in  the  construction 
of  mathematical  instruments,  and  the  skill  with  which  they 
can  be  .manipulated,  that  comparatively  little  trouble  in 
making  observations  arises  from  the  instrument  itself. 

2.  External  Conditions. — The  great  obstacles  to  accu- 
rate work  arise  from  the  influence  of  external  conditions — 
conditions  wholly  beyond  the  observer's  or  instrument- 
maker's  control,  and  whose  effect  can,  in  general,  neither 
be  satisfactorily  computed  nor  certainly  eliminated  by  the 
method  of  observation.  We  have  no  means  of  finding  the 
complex  laws  of  their  action.  Many  of  them  can  be  avoided 
by  not  observing  while  they  operate  in  any  marked  degree. 
Thus,  if  while  an  observer  was  reading  horizontal  angles 
on  a  lofty  station  a  strong  wind  should  spring  up,  it  would 
be  useless  for  him  to  continue  the  work.  If  the  air  com- 
menced to  "boil"  he  should  stop.  If  the  sun  shone  on  one 
side  of  his  instrument  its  adjustments  would  be  so  disturbed 
that  good  work  could  not  be  expected.  So  in  comparisons 
of  standards.  Comparisons  made  in  a  room  subject  to  the 
temperature  variations  of  the  outside  air  would  be  of  little 
value.  The  standards  should  not  only  not  be  exposed  to 
sudden  temperature  changes  during  comparisons,  but  at  no 
other  time  ;  for  it  has  been  shown  by  recent  experiments 
that  the  same  standard  may  have  different  lengths  at  the 
same  temperature  after  exposure  to  wide  ranges  of  tem- 
perature.* 

The  effects  of  external  disturbances  may  sometimes  be 
eliminated,  in  part  at  least,  by  the  method  of  observation. 

*  American  Journal  of  Science,  July,  1881. 


INTRODUCTION.  15 

In  the  measurement  of  horizontal  angles  where  the  instru- 
ment is  placed  on  a  lofty  station,  the  influence  of  the  sun 
causes  the  centre  post  or  tripod  of  the  station  to  twist  in 
one  direction  during  the  day.  When  this  influence  is  re- 
moved at  night  the  twist  is  in  the  opposite  direction.  As- 
suming the  twist  to  act  uniformly,  its  effect  on  the  results 
is  eliminated  by  taking  the  mean  of  the  readings  on  the 
signals  observed  in  order  of  azimuth  and  then  immediately 
in  the  reverse  order. 

Atmospheric  refraction  is  another  case  in  point.  In  ob- 
serving for  time  with  an  astronomical  transit  the  effect  of 
refraction  on  the  mean  of  the  recorded  readings  is  elimi- 
nated by  taking  the  star  on  the  same  number  of  threads  on 
each  side  of  the  middle  thread.  On  the  other  hand,  in  the 
measurement  of  horizontal  angles,  if  long  lines  are  sighted 
over,  or  lines  passing  from  land  over  large  bodies  of  water 
or  over  a  country  much  broken,  the  effects  of  refraction  are 
apt  to  be  very  marked.  As  we  have  no  means  of  eliminat- 
ing the  discordances  arising  in  this  way  by  the  method  of 
observation,  all  we  can  do  is,  while  planning  a  triangulation, 
to  avoid  as  far  as  possible  the  introduction  of  such  lines. 

It  may  happen  that  the  effect  of  the  external  disturbances 
on  the  observations  can  be  computed  approximately  from 
theoretical  considerations  assuming  a  certain  law  of  opera- 
tion. If  the  correction  itself  is  small  this  is  allowable.  As 
an  example  take  the  zenith  telescope,  with  which  the  method 
of  observing  for  latitude  is  such  that  the  correction  for  re- 
fraction is  so  small  that  the  error  of  the  computed  value  is 
not  likely  to  exceed  other  errors  which  enter  into  the  work. 

3.  The  Observer. — Lastly  we  come  to  the  observer  him- 
self as  the  third  element  in  making  an  observation.  Like 
the  external  conditions,  he  is  a  variable  factor  ;  all  new 
observers  certainly  are. 

The  observer,  having  put  his  instrument  in  adjustment 
and  satisfied  himself  that  the  external  conditions  are  favor- 
able, should  not  begin  work  unless  he  considers  that  he  him- 
self is  in  his  normal  condition.  If  he  is  not  in  that  condition 


16  THE   ADJUSTMENT   OF   OBSERVATIONS. 

he  introduces  an  unknown  disturbing  element  unnecessa- 
rily. He  is  also  more  liable  to  make  mistakes  in  his  read- 
ings and  in  his  record.  For  the  same  reason  he  should  not 
continue  a  series  of  observations  too  long  at  one  time,  as 
from  fatigue  the  latter  part  of  his  work  will  not  compare 
favorably  with  the  first.  In  time-determinations,  for  in- 
stance, nothing  is  gained  by  observing  from  dark  until 
daylight. 

The  observer  is  supposed  to  have  no  bias.  A  good  ob- 
server, having  taken  all  possible  precautions  with  the  ad- 
justments of  his  instrument  and  knowing  no  reason  for  not 
doing  good  work,  will  feel  a  certain  amount  of  indifference 
towards  the  results  obtained.  The  man  with  a  theory  to 
substantiate  is  rarely  a  good  observer,  unless,  indeed,  he 
regards  his  theory  as  an  enemy  and  not  as  a  thing  to  be 
fondled  and  petted. 

The  greater  an  observer's  experience  the  more  do  his 
habits  of  observation  become  fixed,  and  the  more  mechanic- 
al does  he  become  in  certain  parts  of  his  work.  But  his 
judgment  may  be  constantly  at  fault.  Thus  with  the  as- 
tronomical transit  he  may  estimate  the  time  of  a  star  cross- 
ing a  wire  in  the  focus  of  the  telescope  invariably  too  soon 
or  invariably  too  late,  according  to  the  nature  of  his  tem- 
perament. If  he  is  doing  comparison  work  involving  mi- 
crometer bisections,  he  may  consider  the  graduation  mark 
sighted  at  to  be  exactly  between  the  centre  wires  of  the 
microscope  when  it  is  constantly  on  the  same  side  of  the 
centre.  This  fixed  peculiarity,  which  none  but  experienced 
observers  have,  is  known  as  their  personal  error. 

In  combining  one  observer's  results  with  those  of  another 
observer  we  must  either  find  by  special  experiment  the  dif- 
ference of  their  personal  errors  and  apply  it  as  a  correction 
to  the  final  result,  or  else  eliminate  it  by  the  method  of  ob- 
servation. Thus  in  longitude  work  the  present  practice 
is  to  eliminate  the  effect  of  personal  error  from  the  final 
result  by  having  the  observers  change  places  at  the  middle 
of  the  work. 


INTRODUCTION.  I/ 

It  is  always  safer  to  eliminate  the  correction  by  the 
method  of  observing  rather  than  by  computing  for  it.  For 
though  it  may  happen  that  so  long  as  instruments  and 
conditions  are  the  same  the  relative  personal  error  of  two 
observers  may  be  constant,  yet  some  apparently  trifling 
change  of  conditions,  such,  for  example,  as  illuminating 
the  wires  of  the  instrument  differently,  may  cause  it  to  be 
altogether  changed  in  character. 

On  account  of  personal  error,  if  for  no  other  reason,  it 
is  evident  that  no  number  of  sets  of  measures  obtained  in 
the  same  way  by  a  single  observer  ought  to  be  taken  as 
furnishing  a  final  determination  of  the  value  of  a  quantity. 
We  must  either  vary  the  form  of  making  the  observations 
or  else  increase  the  number  of  observers,  in  the  hope  that 
personal  error  will  eliminate  itself  in  the  final  combination 
of  the  measures. 

4.  When  all  known  corrections  for  instrument,  for  exter- 
nal conditions,  and  for  peculiarities  of  the  observer  have 
been  applied  to  a  direct  measure,  have  we  obtained  a  cor- 
rect value  of  the  quantity  measured?  That  we  cannot  say. 
If  the  observation  is  repeated  a  number  of  times  with  equal 
care  different  results  will  in  general  be  obtained. 

The  reason  why  the  different  measures  may  be  expected 
to  disagree  with  one  another  has  been  indicated  in  the  pre- 
ceding pages.  There  may  have  been  no  change  in  the  con- 
ditions of  sufficient  importance  to  have  attracted  the  ob- 
server's attention  when  making  the  observations,  but  he 
may  have  handled  his  instrument  differently,  turned  certain 
screws  with  a  more  or  less  delicate  touch,  and  the  exter- 
nal conditions  may  have  been  different.  What  the  real  dis- 
turbing causes  were  he  has  no  means  of  knowing  fully.  If 
he  had  he  could  correct  for  them,  and  so  bring  the  meas- 
ures into  accordance.  Infinite  knowledge  alone  could  do 
this.  With  our  limited  powers  we  must  expect  a  residuum 
of  error  in  our  best  executed  measures,  and,  instead  of  cer- 
tainty in  our  results,  look  only  for  probability. 

The  discrepancies  from  the  true  value  due  to  these  un- 


1 8  THE   ADJUSTMENT   OF   OBSERVATIONS. 

explained  disturbing  causes  we  call  errors.  These  errors 
are  accidental,  being  wholly  beyond  all  our  efforts  to  con- 
trol. As  soon  as  they  are  known  to  be  constant,  or  we  learn 
the  law  of  their  operation,  they  cease  to  be  classed  as 
errors. 

A  very  troublesome  source  of  discrepancies  in  measured 
values  arises  from  mistakes  made  by  the  observer  in  reading 
his  instrument  or  in  recording  his  readings.  Mistakes  from 
imperfect  hearing,  from  transposition  of  figures  and  from 
writing  one  figure  when  another  is  intended,  from  mistak- 
ing one  figure  on  a  graduated  scale  for  another,  as  7  for  9, 
3  for  8,  etc.,  are  not  uncommon.  These  also  must  be  classed 
as  accidental  errors,  theoretically  at  least. 

Having,  therefore,  taken  all  possible  precautions  in  mak- 
ing the  observations  and  applied  all  known  corrections  to 
the  observed  values,  the  resulting  values,  which  we  shall  in 
future  refer  to  as  the  observed  values,  may  be  assumed  to 
contain  only  accidental  errors.  We  are,  then,  brought  face 
to  face  with  the  question,  How  shall  the  value  of  the  quan- 
tity sought  be  found  from  these  different  observed  values? 

Synopsis  of  Mathematical  Principles  Employed. 

For  convenience,  and  in  order  to  avoid  multiplicity  of 
references,  the  leading  principles  of  pure  mathematics  made 
use  of  in  the  further  development  of  our  subject  are  here 
placed  together. 

5.  Probability. — (i)  The  probability  of  the  occurrence  of 
an  event  is  represented  by  the  fraction  whose  denominator  is  tJie 
number  of  possible  occurrences,  all  of  which  are  supposed  to  be 
independent  of  one  another  and  equally  likely  to  happen,  and 
whose  numerator  is  the  number  of  these  occurrences  favorable  to 
the  event  in  question.  Thus  if  an  event  may  happen  in  a  ways 
and  fail  in  b  ways,  all  equally  likely  to  occur,  the  probability 

of  its  happening  is  -    -r—,,  and  of  its  failing       .    , ,  certainty 

tt     L      j  U  dr     *"j  U 

being  represented  by  unity. 


INTRODUCTION.  19 

(2)  If  there  are  n  events  independent  of  each  other,  and  the 
probability  of  the  first  happening  is  if,\,  of  the  second  ipt,  and  so 
on,  the  probability  that  all  will  happen  is 


Thus  if  an  urn  contains  two  white  and  seven  black  balls 
the  probability  of  drawing-  white  at  each  of  the  first  two 
trials,  the  ball  not  being  replaced  before  the  second  trial,  is 

2          I  _      I 

9X  8-^6 
6.  Definite    Integrals.  —  (a)  To  find    the    value    of 


Let  /  =:  A'-sr  /  then  regarding  x  as  a  constant  in  integrat- 
ing, we  have 


~f*dt  =    r~f**xdz 

\J     0 

Multiply    each    member   by   /   e~*dx,    which    is    equal   to 

J  o 

/e^dt,  since  the  limits  of  integration  are  the  same  ;  then 
• 

I  y^.*  \      1  f**>  /-.» 

/   g+dt\   -   I  ds    I    <r**+* 

(      ^/     0  )  v/     0  J     0 


Hence   f  e^dt  =  |  yC 

J    0 

(b)  To  find  the  value  ol    f 


20  THE  ADJUSTMENT  OF  OBSERVATIONS. 

Integrating  by  parts,  we  have 

/°°  /        fs>-aW\°°  r*      traW 

aff*»dt  =  (-*!—-}      +/       L  _  d(af) 
\       2a  /-»    ^  J  -»      2a*     ^    ' 


-  20* 

Also,  similarly, 


(c)  To  find  the  value  of 

The  value  of  this  integral  cannot  be  expressed  exactly 
in  a  finite  form,  but  may  be  found  approximately  as  fol- 
lows : 

Expanding  e'*  in  a  series  and  integrating  each  term 
separately,  we  have 


/ 


dt 


a3    .      l  a* 
=  a  ---  ---- 

3         1.2  5 


This  series  is  convergent  for  all  values  of  a;   but  the  con- 
vergence is  only  rapid  enough  for  small  values  of  a. 

For  large  values  of  a  it  is  better  to  proceed  as  follows: 
Integrating  by  parts, 


fe-*dt=  f-^-de^ 

J  J         2t 

i    _<2      i     /V  ,. 

=  --  a  *   --    /    —  T  dt 
2t  2.J? 


Hence 


J  a 


^  ~~= 


INTRODUCTION.  21 

But 

/a  S**  f^* 

e-*dt  --  I    e~*dt  —  I    e^dt 
J    o  *J     a 

VTT         r" 
'~~J *£   dt 
.  • .  finally, 

J   e"dt  =  ~?    '  ^T  I l  ~~  i  +  ^y  ~ Sf  +  •  •  •    | 

It  is  easily  shown  that,  by  stopping  the  summation  at 
any  term,  the  result  will  differ  from  the  true  value  by  less 
than  the  term  stopped  at. 

/(L 
e'^dt  may  be 

computed    from    the    above    formulas    for   any    numerical 
value  of  a. 

7.  Taylor's  Theorem. — (a)  If  /  (x)  is  any  function  of 
x,a.ndf(x-{-/i)  is  to  be  developed  in  ascending  powers  of//, 
then 

(O 


A    more    rapid    approximation    is    obtained    by  putting 
the  development  in  the  following  form  : 


By  subtraction  and  transposition, 
/(«  +  A)  =  /  W  +  /< 


22  THE   ADJUSTMENT   OF   OBSERVATIONS. 

(b)  Let  F  denote  a  function  of  a  series  of  quantities  X, 
F,  .  .  .  expressed  by  the  relation 

F=f(X,  F,  .  .  .  ) 

and  let  X',  Y',  .  .  .  denote  approximate  values  of  X,  Y, 
.  .  .  and  x,  y,  .  .  .  the  corrections  to  these  approximate 
values,  so  that 

X—X'+x 

Y=Y'+y 


then 


6'F    ,,  d'F  ,    d'F    , 

'         ' 


where  —  —  »    —  —  •    .   .  .  are  the  values  found  by  diflferenti- 
6X'      6  Y' 

ating  f  (X,  Y,  ...),...  with  respect  to   X,  F,  .  .  .  and 
then  substituting  X',  Y',  .  .  .  for  X,  F.  .  .  . 

If  the  corrections  x,  y,  .  .  .  are  so  small  that  their 
squares  and  higher  powers  may  be  neglected  and  they  are 
written  dX1  ,  dY't  .  .  .  and  F  —  f  (X',  Y',  .  .  .)  is  written 
dF,  then 


which  is  exactly  the  result  found  by  differentiating  F,  which 
is  a  function  of  X,  Y,  .  .  .  with  respect  to  these  quanti- 
ties. 

We  shall  use  one  form  or  the  other  as  may  be  most  con- 
venient. 

Ex.  i.   If  -v  is  a  very  small  correction  to  the  number  N,  required  to  express 
log  (N+v)  in  the  linear  form. 


INTRODUCTION.  23 

We  have 

\og(W+v)  =log^+  ^ 
dN 


where  mod.  is  the  modulus  of  the  common  system  of  logarithms. 
With  a  seven-place  table  we  may  use  the  formula 

log  (N+  v)  =  log  N  +  5N  v 

where  <5N   is  the  tabular  difference  corresponding  to  one  unit  for  the  number. 

For  small  numbers,  however,  it  is  better  to  take  <5N   from  the  table  for  the 
form 


Thus  from  the  table 

log  (6543.2  +  ^)  =  3.8157902  +  0.0000664  v 
log  (654.32  +  v)  —  2.8157902  +  0.0006637  v 
log  (65.432  +  ^)  =  1.8157902  +  0.0066378  v 

Ex,  2.  In  a  ten-place  log.  table  where  angles  are  given  at  regular  intervals, 
required  to  find  log  sin  (A+a)  when  A  is  given  in  the  table  and  a  is  a  num- 
ber of  seconds  less  than  the  tabular  interval. 

We  have 

log  sin  (A  +a)=\og  sin  A  +-T-J  1  log  sin  (A  +  -  )  [  a 

=log  sin  A  +  mod.  sin   i"   a  cot   (A  H  —  ] 
Now, 

log  mod.  =:  9.6377843  —  10 
log  sin  i"  =  4.6855749  —  10 
log  io7  =  7. 


1.3233592 

which  expresses  log  (mod.  sin  i")  in  terms  of  the  seventh  place  of  decimals  as 
the  unit.     Hence 

log  sin  (A  +  a) 

=  log  sin  ,4+log-'  j  i.3233592  +  logrt+  (log  cot  A  +  -  X  diff.  for  i"j  j- 

With  Vega's  Thesaurus,*  which  gives  the  log.  functions  to  single  seconds 
to  ttie  end  of  the  first  degree  and  afterwards  for  every  io",  log  sin  (A  +a)  can  be 
found  from  the  above  expression  for  values  of  A  >  3°,  and  also  when//  lies 
between  20'  and  2°  to  within  less  than  unity  in  the  tenth  decimal  place.  Be- 
tween 2°  and  3°  the  difference  may  be  as  large  as  3  units  in  the  tenth  phice 
from  the  value  found  by  carrying  out  the  formula  more  exactly.  But  in  the 

*  Thesaurus  Logaritkmorum  Contpletus.    Lipsiae,  1794. 


24  THE  ADJUSTMENT   OF   OBSERVATIONS. 

Thesaurus,  in  the  trigonometrical  part,  "  the  uncertainty  of  the  last  figure 
amounts  to  4  units/'*  Hence  the  above  process  is  in  general  sufficient  when 
this  table  is  used. 

With  a  seven-place  table  of  log.  sines  it  would  be,  in  general,  sufficient  to 
take  the  tabular  difference  <5A  for  i"  for  the  angle  A  as  the  value  of 


TT    1     10§ 

dA.    ( 
so  that 

log  sin  (A  +a)=log  sin  A  +  6A  a. 

Ex.  3.  If  A  is  the  approximate  value  of  an  angle,  and  v  a  correction  to  it 
so  small  that  its  square  and  higher  powers  may  be  neglected,  required  to  ex- 
press log  sin  (A  +  v)  in  the  linear  form,  using  a  ten-place  table. 

Let  AI  be  the  angle  nearest  to  A  in  the  table,  and  set 

then 

log  sin  (A  +v) 

=log  sin  (A i  +  a  +  v) 

(  a\ 

=log  sin  AI  +  mod.  sin    i"   a  cot  (   AI  +  -  \  +  mod.  sin  i"  cot  (A\  +  a)  v 

—  log  sin  AI  +  log"1  1 1.3233592  +  log  a  +  (log  cot  AI  +-  X  diff.  for  i")  £ 

+  log-1  1 1.3233592  +  (log  cot  A  i  +  a  X  diff.  for  i")|  v 
Expand  log  sin  (68°  16'  3: 


1.3233592 

log  COt  A!  9.6003780 


A!—  68°  16'  30" 

a  =  2".  076 


0.9237372 

«Xdiff.  i"          —127 


0.9237245         8,3893 


-?Xdiff.  i"  +64 

2 


0.9237309 

logtf  0.3172273 

1.2409582  I7,4l6 

log  sin  68°  16'  30"  9.9680022,271 

Hence  log  sin  (68°  16'  32" .076  +  v)=          9.9680039,687  +  8,3893  v 

when  the  difference  is  expressed  in  terms  of  the  seventh  decimal  place  as  the 
unit. 

*  Bremiker's  edition  of  Vega,  translated  by  Fischer.     Preface,  p.  10. 


INTRODUCTION.  2' 

With  a  seven-place  table,  except  for  small  angles  or  angles  near  180° 
it  will  be  sufficient  to  take 


log  sin  (A  i 


=^o    sn 


when  6\   is  the  tabular  difference  corresponding  to  i"  for  the  angle  A\.     It 
can  be  taken  by  inspection  from  the  table.     Thus, 

log  sin  (68°  16'  32"  +  v)=().  9680039  +  8,4  v 

8.  Interpolation.  —  So  far  as  interpolation  is  concerned, 
we  have  mainly  to  deal  with  the  logarithms  of  trigonometric 
functions.  The  differences  between  the  successive  values 
given  in  a  table  are  first  differences,  and  the  differences  be- 
tween the  successive  first  differences  are  second  differences. 
Beyond  second  differences  we  do  not  need  to  go. 

This  may  be  expressed  in  tabular  form  : 


Function. 

First  diff. 

Second  diff. 

f(A) 

f(A+a) 

4 

f(A+2a) 

Hence 


f(A+a)    := 


Generally 

f(A  +  na)  =f(A) 
which  is  Newton's  formula. 


Ex.  In  a  ten-place  table  of  log.  sines  in  which  values  arc 


26  THE   ADJUSTMENT   OF   OBSERVATIONS. 

given  to  every  10  seconds,  required  log  sin  A  when  A  is 
any  angle. 

Let  Al=  the  part  of  the  given   angled  to  the   nearest  10 

seconds  that  occurs  in  the  table. 
«=ithe  units  of  seconds  and   parts  of  seconds  in  the 

given  angle. 
d^  d^=i  the  first  and  second  tabular  differences. 

then 


•    log  sin  A  =  log  sin  f^,-j  ---  .  ioj 


=  logsin  AA-a       +  tfaho  —  a)-*-.         (i) 

10  100 

Writing  this  in  the  form, 

log  sin  ,4  =  log  sin  ,4  ,+  «   j  A  -f  (5  -^\  A  i    (2) 

we  have  the  convenient  rule  :  Assume  the  second  difference 
constant  throughout  the  interval  a.  Then  from  the  first 
and  second  differences  find  by  simple  interpolation  the 
value  of  the  difference  at  the  middle  of  the  interval.  This 
difference  multiplied  by  the  interval  gives  the  correction  to 
the  tabular  log.  sine. 

To  find  log  sin  68°  16'  32".O76  : 
From  the  table, 

log  sin  68°  i6'3o"=     9.9680022,271 

di=  83,889  for  10"  in  units  of  the  seventh  decimal  place 
d%=     0,012 

Hence,  from  equation  (i), 

Corr.  to  tab.  value  =2.076X8,  3889  +  ^^  (10  —  2.076)  — 

2  100 

17,416 


.'.  log  sin  68°  16'  32". 076  =  9.9680039,687 


INTRODUCTION.  2J 


The  difference  at  35"=  8,3889. 

We  want  it  at  31". 038,  the  middle  of  the  interval. 

Now,  change  of  first  difference  =  0,00012  for  i". 

Hence  corr.  to  first  difference  =  (35  —  31.038)  X  0,00012 

=  0.0005 
First  diff.  =  8,3889 


Diff.  required  =  8,3894 
And  2.076  X  8,3894—  17,416  as  before. 

9.  Periodic  Series.—  To  sum  the  series 

Cos  o  -\-  cos  8  -\-  cos  2  0  -f-    .  .  .  -f-  cos  (;/  —  \)0 
Sin  (?-(-sin  #-|-sin  20-\-    .  .   .   -|-sin  (;/  —  1)6 

where  8=~  —  ,  n  being  an   integer,  we  may  proceed  as  fol- 

lows : 

If  6  is  the  angle  which  a  line  B  'OB  makes  with  OA  then 
the  projection  of  OB  on  OA  is  OB  cos  0,  and  the  projec- 
tion of  OB'  is  —OB  cos  6  if  OB'=  OB.  The  projections  on 
a  line  at  right  angles  to  OA  are  OB  sin  0  and  —  OB  sin  6 
respectively. 

If  we  divide  the  circumference  of  a  circle  into  n  equal 
parts  at  the  points  A,  B,  .  .  .  then  each  angle  at  the  centre 

^60° 

O  is  ^  —  ,  or  6,  and  by  projecting  the  lines  OA,  OB,  ...  on 
n 

the  diameter  through  A  we  find  the  sum  of  the  first  series 
to  be  zero,  and  by  projecting  the  same  lines  on  the  diameter 
perpendicular  to  OA  we  find  the  sum  of  the  second  series 
to  be  also  zero. 

These  results  may  be  written  : 


-  cos  ;//    =  o  -   sn  m    = 

where  m  assumes  all  values  from  o  to  n  —  I. 
Hence  it  follows  that 

-  sin  in0  cos  wti=  l/2  -  sin  in  2^  — 


v  n    i    ,  '  v  u         n 

_  cos   »i0  h/2-  cos  ;//  20  = 

~>  2 

-  sin3  m0  -  —  l/2-  cos  ///  20  - 


28  THE   ADJUSTMENT   OF   OBSERVATIONS. 

10.  Notation.  —  The  following  convenient  notation,  intro- 
duced by  Gauss,  is  now  very  generally  used  in  the  method 
of  least  squares. 

If  «„  a,2,  .  .  .  are  quantities  of  the  same  kind,  their  alge- 
braic sum  is  denoted  by  [a],  and  the  sum  of  their  squares  by 
[aa\  or  [V],  so  that 


[aa\  or  [a*]  =  a?  +  a*  +  ...      +  a\ 
Also, 

ab~  =  ab--a--  .  ,  .  a  nbn 


—    j_  _i_     **    i  i     »n 

L  c  J  c,  c^  cn 

We  shall  use  the  symbol  [a  to  denote  the  sum  of  a  series 
of  quantities  a  all  taken  with  the  same  sign. 


CHAPTER    II. 

THE   LAW    OF   ERROR. 
The    Arithmetic    Mean. 

ii.  (a)  When  the  quantity  measured,  is  the  quan- 

tity to  be  found.  —  If  Mlt  Mt  .  .  .  Mn  are  n  direct  and  in- 
dependent measures  of  a  quantity,  T,  we  may  write 

T-M=A 


where  J,,  Av  .  .  .  An  indicate  the  differences  between  T  and 
the  observed  values,  and  are  therefore  the  errors  of  obser- 
vation. 

We  have  here  n  equations  and  n  -f-  I  unknowns.  What 
principle  shall  we  call  to  our  aid  to  solve  these  equations 
and  so  find  T,  A^  A^  .  .  .  JB?  In  answering  this  question 
I  shall  follow  the  order  of  natural  development  of  the  sub. 
ject,  which,  in  the  main,  is  also  the  order  of  its  historical 
development. 

The  value  sought  must  be  some  function  of  the  ob- 
served values  and  fall  between  the  largest  and  smallest  of 
them.  If  the  observed  values  are  arranged  according  to 
their  magnitudes  the}'  will  be  found  to  cluster  around  a 
central  value.  On  first  thoughts  the  value  that  would  be 
chosen  as  the  value  of  T  would  be  the  central  value  in  this 
arrangement  if  the  number  of  observations  were  odd,  and 
either  of  the  two  central  values  if  the  number  were  even. 
In  other  words,  a  plausible  value  of  the  unknown  would  be 
that  observed  value  which  had  as  many  observed  values 
greater  than  it  as  it  had  less  than  it.  Now,  since  a  small 
change  in  any  of  the  observed  values,  other  than  the  central 


3O  THE   ADJUSTMENT   OF   OBSERVATIONS. 

value,  would  in  general  produce  no  change  in  the  result, 
the  number  of  observations  remaining  the  same,  this  method 
of  proceeding  might  be  regarded  as  giving  a  plausible  re- 
sult, more  especially  if  the  observed  values  were  widely 
discrepant. 

On  the  other  hand,  the  taking  of  the  central  value  is  ob- 
jectionable, because  it  gives  the  preference  to  a  single  one 
of  the  observed  values,  while  if  these  values  are  supposed 
to  be  equally  worth)'  of  confidence,  as  it  is  reasonable  to 
take  them  in  the  absence  of  all  knowledge  to  the  contrary, 
each  ought  to  exert  an  equal  influence  on  the  result.  We 
may,  therefore,  with  more  reason  assume  the  value  of  T  to 
be  a  symmetrical  function  of  the  observed  values.*  Also, 
since  a  change  in  the  number  of  observations  should  pro- 
duce no  change  in  the  form  of  this  function,  it  follows  that 
the  function  must  be  of  such  a  form  as  to  satisfy  the  condi- 
tion that  when  the  observed  values  are  all  equal  to  one 
another  it  will  reduce  to  this  common  value  ;  that  is,  if 
T=f(M1M,  .  .  .  Mn],  and  M,=  Ma=  .  .  .  Mn  =  M,  then 
f(M,M,  .  .  .  M)  =  M.  For  if  we  had  a  single  observa- 
tion, then  necessarily  f(M)  =  M. 

Let,  then,  V  be  a  symmetrical  function  of  Ml  M^  .   .  .  Mn> 
and  put 


Expanding  by  Taylor's  theorem  : 


=  f(V,V,.  .  .)-\v\ 

where  //,  K,  .   .   .  .  are  terms  involving  the  second,  third, 
.   .   .  powers  of  the  small  quantities,  vlt  v«,  .  .  .  vn 

*  See  Reuschle,  Crelle  Jour.  Math.,  vol.  xxvi.  ;  Schiaparelli,  Rendiconti  del  R.  Insti- 
tute Lombards,  1868;  Astron.  Nachr.,  2068,2097;  Stone,  Month.  Not.  Roy.  Astron.  Soc., 
vol.  xxxiii.  ;  Astron.  Nachr.,  2092;  Ferrero,  Expos,  del  Met.  dei  Min.  Quacfr.,  Florence, 
1876.  Also  Fechner,  Ueber  den  Ausgangswerth  der  kl.  Abweichungssumme,  Leipzig,  1874. 


THE    LAW    Of   ERROR.  31 

The  simplest  symmetrical  function  of  the  observed  val- 
ues that  can  be  chosen  as  the  form  for  Fis  their  arithmetic 

mean  —  that  is,  '-  —  ^.     If  we  take  V  equal  to  this  value,  then 
n 

from  equation  (i)  by  addition  [?>\—o,  and,  neglecting  powers 
of  v  higher  than  the  first,  equation  (3)  is  satisfied  identically. 
Thus  far,  therefore,  it  would  appear  that  the  arithmetic 
mean  may  be  taken  as  one  solution.  It  may  happen  that  the 
values  M1}  Mv  .  .  ,  J/M  are  of  such  a  nature  that  some  other 
symmetrical  function  than  the  arithmetic  mean  will  satisfy 
(2)  better  than  will  the  arithmetic  mean.  That  the  arith- 
metic mean  is  on  the  whole  the  best  form  for  the  function 
f(Mlt  Mv  .  .  .  Mn],  when  Mlt  Mv  .  .  .  Mn  are  direct  meas- 
ures of  some  phenomenon  in  the  sciences  of  observation, 
which  sciences  only  we  intend  to  consider,  may  be  confirm- 
ed by  a  comparison  of  results  flowing  from  this  hypothesis 
with  the  records  of  experience.  This  we  shall  do  later  (see 
Art.  24,  37,  51). 

The  older  mathematicians,  as  Cotes  and  Simpson,  laid 
the  foundations  of  our  subject  by  announcing  the  principle 
of  the  arithmetic  mean.  Gauss,  to  whom  we  owe  the  first 
complete  exposition,  assumed  the  arithmetic  mean  as  a 
plausible  hypothesis;*  and  Hansen,  who  made  the  next 
great  advances,  started  from  it  as  an  axiom.  The  princi- 
ple itself  may  be  stated  as  follows  :  If  we  have  n  observed 
values  of  an  unknown,  all  equally  good  so  far  as  we  know, 
the  most  plausible  value  of  the  unknown  (best  value  on  the 
whole)  is  the  arithmetic  mean  of  the  observed  values. 

12.  By  adding  equations  (i),  Art.  n,and  taking  the  mean, 
we  have 


V  .. 

—    *       \^  ' 

n 

The  last   term   of  this  equation  will   become  very  small  if, 

*  Gauss'    words   are,  "  Axiomatis  loco  haberi  solet  hypothesis."     (Theoria  Motus,  lib.  z, 
sec.  3.) 


32  THE   ADJUSTMENT   OF   OBSERVATIONS. 

n  being  very  large,  the  sum  [J]  of  the  errors  remains 
small.  Now,  if,  after  making  one  observation  and  before 
making  another,  we  readjust  our  instrument,  determine 
anew  its  corrections,  choose  the  most  favorable  conditions 
for  observing,  and  vary  the  form  of  procedure  as  much  as 
possible,  it  is  reasonable  to  suppose  that  the  disturbing  in- 
fluences will  balance  one  another  in  the  result,  following 
from  the  proper  combination  of  the  observed  values.  It 
may  take  an  infinite  number  of  trials  to  bring  this  about. 
In  the  absence  of  all  knowledge  we  cannot  say  that  it  will 
take  less.  And,  reckoning  mistakes  in  reading  the  instru- 
ment or  in  recording  the  readings  as  accidental  errors,  an 
infinity  of  a  higher  order  than  the  first  may  be  required  to 
eliminate  them. 

In  other  words,  there  being  no  reason  to  suppose  that 
an  error  in  excess  (or  positive  error)  is  more  likely  to  occur 
on  the  whole  than  an  error  in  defect  (or  negative  error),  we 

may,  when  n  is  a  very  large  number,  consider  LJ  to  be  an 

n 

infinitesimal  with  respect  to  T.  We  may,  therefore,  in  this 
case  put 

V=  T 

—  that  is,  when  the  number  of  observed  values  is  very  great  the 
arithmetic  mean  is  the  true  value. 

13.  From  the  principle  of  the  arithmetic  mean  two  im- 
portant inferences  may  be  derived.  For,  taking  the  arith- 
metic mean,  V,  of  n  observed  values  of  an  unknown  as  the 
most  plausible  value  of  that  unknown,  we  may  write  our 
observation-equations  in  the  form, 


V-Mn=vn 

where  v^  t/a,  .  .  .  vn  are  called  the  residual  errors  of  observa- 
tion, or  simply  the  residuals. 


THE   LAW   OF   ERROR.  33 

(a)  By  addition 

nV  -[M~\  =  [v] 

and  .  •. 

[>]=0.  (2) 

—  that  is,  the  sum  of  the  residuals  is  zero  ;  in  other  words,  the 
sum  of  the  positive  residuals  is  equal  to  the  sum  of  the  negative 
residuals. 

There  is  a  very  marked  correspondence  between  the 
series  in  which  n  is  infinitely  great  and  T  is  the  true  value, 
and  a  series  in  which  n  is  finite  and  the  arithmetic  mean  V 
is  taken  as  the  best  value  attainable.  For  in  the  first  case 
the  sum  of  the  errors,  J,  divided  by  ;/,  is  zero,  and  in  the  sec- 
ond the  sum  of  the  residuals,  v,  is  zero. 

(b)  Let  V  be  any  assumed  value  of  the  unknown  other 
than  the  arithmetic  mean,  and  put 


V'  —  M  —  T'' 

(3) 


From  equations  (i)  and  (3),  by  squaring  and  adding, 

[vv]  =  nVV-2V  [M]  +  [MM] 
[v'vf]  =  n  V  V-  2  V  [M]  +  [MM] 

Hence  by  a  simple  reduction, 


[t^=[W]  +  *(r'         ") 

Now,  |F'    -t  —  J  J  ,  being  a  complete  square,  is  always 

positive. 

.'.  [W]>  [>'?'] 

—that  is,  the  sum  of  the  squares  of  tlie  residuals  v,  found  by 
taking  the  arithmetic  mean,  is  a  minimum. 

Hence  the  name  Method  of  Least  Squares,  which  was  first 
given  by  Legendre. 

Let  us  recapitulate  the  three  forms  of  solution  proposed 


34  THE  ADJUSTMENT   OF   OBSERVATIONS. 

for  finding  the  most  plausible  value  Fof  the  unknown  from 
the  n  equations  : 


(1)  Find  all  possible  values  of  Fand  take  the  mean.     The 
values  of  Fare  M^  Mv  .  .  .  Mn,  since  for  each  observation 
considered  singly  the  best  value  must  be  the   directly  ob- 

served one,  and  the  mean  of  these  values  is  !=  —  J 

n 

(2)  Solve  simultaneously,  making  [v~\  —  O.    This  gives 


or 

n 

the  arithmetic  mean. 

(3)  Solve   simultaneously,    making    [w]  =  a    minimum. 
This  gives 

(V-M$+(V-M$+  .  .   .  (V-  Mn)*  =  a  min. 
Differentiating  with  respect  to  F, 


or 

n 

the  arithmetic  mean. 

14.  When  the  quantity  measured  is  a  function 
of  the  quantities  to  be  found.  —  We  pass  now  to  the 
more  general  but  equally  common  case  in  which  the  ob- 
servations, instead  of  being  made  directly  on  the  quantity 
to  be  determined,  are  made  indirectly  —  that  is,  made  on  a 
quantity  which  is  a  function  of  the  quantities  whose  values 
are  to  be  found. 


THE   LAW  OF  ERROR.  35 

Thus,  let  the  function  connecting  the  observed  quantity 
Tand  the  unknowns  X,  Y,  .  .  .  be 

T=f(X,  F,  .  .  .  )  (i) 

in  which   the  constants  involved  are  given  by  theory  for 
each  observation. 

If  Mlt  Mt,  .  .  .  are  the  observed  values  of  71,  the  equa- 
tions for  finding  the  unknowns,  when  reduced  to  the  linear 
form,  may  be  written 


(2) 


in  which  alt  &lt  ...  Z,,  .  .  .  are  known  constants,  and  vlt 
vv  .  .  .  are  the  residual  errors  of  observation. 

Now,  if  the  number  of  equations,  «,  is  equal  to  the  num- 
ber of  unknowns,  »,-,  the  values  of  X,  F,  .  .  .  may  be  found 
by  the  ordinary  algebraic  methods,  and  if  substituted  in  the 
equations  will  satisfy  them  exactly.  But  if  the  number  of 
equations  exceeds  the  number  of  unknowns,  the  values 
found  from  a  sufficient  number  of  the  equations  will  not 
in  general  satisfy  the  remaining  equations  exactly.  Many 
such  sets  of  values  may  be  found,  which  are  therefore  all 
possible  solutions.  But  of  all  these  possible  sets  some  one 
will  satisfy  the  equations  better  than  any  of  the  others.  We 
have  so  far  no  means  of  knowing  when  we  have  found  this 
most  plausible  (best  on  the  whole)  set  of  values.  With  a 
single  unknown  the  arithmetic  mean  gives  the  most  plausi- 
ble result.  Let  us  see  if  a  method  of  finding  means  corre- 
sponding to  those  of  Art.  13  will  apply  to  these  equations. 

For  simplicity  in  writing  take  the  three  equations: 


where  alt  b^  .  .  .  are  known,  and  X,  Fare  to  be  found. 
6 


36  THE  ADJUSTMENT  OF  OBSERVATIONS. 

(i)  Find  all  possible  values  of  X  and  Y,  and  combine 
them. 

To  do  this  we  form  all  possible  sets  of  two  equations 
and  solve  each  set.  Thus, 


whence  at  once 

(a  A  -  a  A}  X=  b,  M,  -  b,M, 
(afa  —  «,£,)  Y—  0,Ma  —  a,M, 

(a,d3  —  a3d^  X=  btM^  —  b^M3 
(ajbt  —  a&)  Y=  a,M3  —  a3M, 

(aj),  —  a,b,)  X=  b3M,  —  b,M, 
(a&  —  a&)  Y—  a,M3  —  a3Mt. 

In  combining  these  values  of  Jf  and  of  Y  we  are  met  by 
a  difficulty.  It  would  not  do  to  take  the  arithmetic  means 
as  the  most  plausible  values,  for  X  and  Fmay  be  better  de- 
termined from  one  set  of  equations  than  from  another,  and 
the  arithmetic  mean  gives  the  most  plausible  value  only  on 
the  assumption  that  all  of  the  values  combined  in  it  are  of 
equal  quality.  It  is  necessary,  therefore,  to  have  a  method 
of  combining  observations  of  different  quality  before  we  can 
find  X  and  Fin  this  way. 

(2)  The  simultaneous  solution  of  the  equations  by  mak- 
ing the  sum  of  the  residuals  equal  to  zero. 

Hence  X,  Y  should  be  found  from 


which  is  impossible,  as  this  equation  may  be  satisfied  by  an 
infinite  number  of  values  of  X  and  Y.  The  principle  is, 
therefore,  insufficient. 

(3)  The  simultaneous  solution  by  making  the  sum  of  the 
squares  of  the  residuals  a  minimum.     We  have 

(a,X  '-\-b.Y-  My  +  ...  +  ...  —a  rain. 


THE   LAW   OF    ERROR.  37 

Differentiating  with  respect  to  X,  Y  as  independent  vari- 
ables, we  find 

\_aa\X-\-\ab\  Y=[aM~\ 
[ad]  X+  [bb\  Y=  [bM] 
where 

[aa\  =  a, a,  -f-  aji,  -f  a3a3 
[ab]  =  ajbi  -j-  a&  +  aj, 

This  method  gives  as  many  equations  as  unknowns,  and  so 
but  one  set  of  values  of  X  and  Fcan  be  found. 

We  cannot,  however,  say  that  we  have  found  the  most 
plausible  values  of  X  and  Y.  All  we  can  say  is  that  the  last 
method  employed  reduces  the  number  of  equations  to  the 
number  of  unknowns  and  gives  us  one  set  of  values  of  X 
and  Y,  and  that  the  same  principle  applied  to  the  special 
case  of  one  unknown  gives  the  most  plausible  value  of  that 
unknown,  in  that  it  gives  the  arithmetic  mean.  Analogy, 
however,  would  lead  us  to  suspect  that  we  have  found  the 
most  plausible  values  of  X  and  Y,  With  one  unknown,  if 
the  separate  observed  values,  represented  by  lines  of  the 
proper  length,  are  plotted  along  a  straight  line  from  a  cer- 
tain assumed  origin,  and  equal  weights  are  placed  at  the  end 
points,  the  position  of  the  centre  of  gravity  of  the  weights 
will  coincide  with  the  end  of  the  line  representing  the 
arithmetic  mean  of  the  distances,  and  the  sum  of  the  squares 
of  the  distances  of  the  weights  from  the  centre  of  gravity  is 
a  minimum. 

Again,  the  centre  of  gravity  of  n  equally  well-observed 
positions  of  a  point  in  space  would  be  the  most  plausible 
mean  position  to  take  for  the  point.  But  it  is  a  well-known 
principle  that,  if  equal  weights  be  placed  at  «  points  in 
space,  the  centre  of  gravity  of  these  weights  is  at  a  point 
such  that  the  sum  of  the  squares  of  its  distances  from  the 
weights  is  a  minimum.*  On  this  principle  Legendre  found- 
ed the  rule  of  minimum  squares,  and  he  employed  the  rule 

*  See,  for  example,  Todhunter's  Statics^  Art.  138. 


21101)5 


38  THE  ADJUSTMENT  OF  OBSERVATIONS. 

as  giving  a  convenient  method  of  solution  in  the  class  of  prob- 
lems under  consideration. 

The  Law  of  Error  of  a  Single  Observed  Quantity. 

15.  With  a  single  unknown  we  have  seen  that  the  most 
plausible  value  is  the  arithmetic  mean  of  the  independently 
observed  values,  and  that  it  can  be  found  by  making  the 
sum  of  the  squares  of  the  residuals  a  minimum.  The 
methods  are  equivalent. 

With  more  than  one  unknown  we  have  failed  to  find  this 
correspondence  of  methods.  The  reasoning  from  analogy 
in  the  preceding  article  is  well  enough  as  far  as  it  goes,  but 
it  is  not  conclusive.  The  difficulty  lies  in  combining  values 
not  equally  good.  We  must,  therefore,  devise  some  method 
of  combining  such  values  before  a  rule  for  finding  means 
can  be  applied  to  several  unknowns. 

Now,  when  several  independent  measures  of  the  same 
quantity,  all  equally  good,  have  been  made,  it  must  be  grant- 
ed that  errors  in  excess  and  errors  in  defect  are  equally 
likely  to  occur  to  the  same  amount  —  that  is,  are  equally 
probable.  Experience  shows  that  in  any  well-made  series 
of  observations  small  errors  are  likely  to  occur  more  fre- 
quently than  large  ones,  and  that  there  is  a  limit  to  the 
magnitude  of  the  error  to  be  expected.  If,  therefore,  a  de- 
notes this  limit  or  maximum  error,  we  must  consider  all  the 
errors  of  the  series  to  be  ranged  between  -\-a  and  —  a,  but 
to  be  most  numerous  in  the  neighborhood  of  zero.  Hence 
the  probability  of  the  occurrence  of  an  error  may  be  as- 
sumed to  be  a  certain  function  of  the  error. 

If,  then,  the  probability  that  an  error  does  not  exceed 
A  be  denoted  by  ?(J),  the  probability  of  an  error  between 
J  and  A-\-dA  is 


—  tp(A)dd    suppose.         (i) 

The  function  <p(A)  is  called  the  law  of  distribution  of  error, 
or  simply  the  law  of  error. 


THE   LAW   OF    ERROR.  39 

The  probability  that  an  error  falls  between  any  assigned 
limits  b  and  a  is  the  sum  of  the  probabilities  <f(J)  dA  ex- 
tending from  b  to  a,  and  is  expressed  in  the  ordinary  nota- 
tion of  the  integral  calculus  by 

A  (2) 

Hence  it  follows  that  the  probability  that  an  error  does  not 
exceed  the  value  a  is 

(3) 

The  properties  of  errors  stated  above  are  not  sufficient 
to  determine  the  form  of  the  function  ^(J)  definitely. 
Among  other  forms  that  might  be  chosen  to  satisfy  them 
the  simplest  would  be 


(4) 

where  h  is  a  constant. 

This  was,  indeed,  that  selected  by  Laplace  in  his  inves- 
tigation Determiner  le  milieu  que  I' on  doit  pr entire  entre  f rot's 
observations  donnees  d'un  meme phtnomene  (Mem.  Acad.  Paris, 
1774).  From  this  form  may  be  readily  derived  the  results 
stated  in  Art.  n. 

The  form  of  <p(J)  may,  however,  be  more  satisfactorily 
determined  by  calling  in  the  aid  of  the  calculus  of  probabil- 
ities. For  if  ^>(4)  dJv  9>(4i)  ^4i>  •  •  •  denote  the  prob- 
abilities of  the  occurrence  of  errors  in  n  observations  be- 
tween J,  and  Jj-j-^/Jj,  4,  and  A^-\-dAv  .  .  .  respectively, 
the  probability  of  the  simultaneous  occurrence  of  this  sys- 
tem of  errors  is  proportional  to  the  product  (see  Art.  5). 

$0(4)  ^(4,)  .  .  .  9r(4«) 
Denote  this  expression  by  </?,  so  that 

Now,  the  true  value   T,  and  therefore  the  values  of  J,,  J,, 
.  .  .  Jw,  are  unknown.     If  we  make  the  expression  for  </>  a 


40  THE  ADJUSTMENT  OF  OBSERVATIONS. 

maximum,  we  should  find  the  most  probable  value  of  the 
unknown.  But  we  have  seen  that  the  most  plausible  value 
of  the  unknown  is  the  arithmetic  mean  of  the  observed  val- 
ues, and  that  when  the  number  of  observations  is  very  large 
the  arithmetic  mean  is  the  true  value  T.  Calling,  then,  the 
most  plausible  value  the  most  probable  value,  we  have, 
when  n  is  large,  the  true  value  by  making  </>  a  maximum. 
The  forfh  of  the  function  <p  will,  therefore,  result  from  this 
hypothesis. 

Now,  since  log  tp  varies  as  </>,  we  must  have  log  </>  a  maxi- 
mum, and  therefore  by  differentiation 


<t>_b  log  y(4)  v/J.        b  log 
dT  <>4      "  dT  '          K       "  dT  ~ 

or 

6  log  y>(4 


„  —  i  ffrL  _  ^      IU^  n^;  ,  j  j  lus  »w  .  ,         /g\ 

—      /          i  *y^ ^i  .^       A  /f  I  3  A      f\  A 

since  from  equation  (i),  Art.  11, 

^4       ^/4  ^4 


dT~dT  ~dT 

But,  from  the  principle  of  the  arithmetic  mean,  when  the 
number  of  observations  is  very  great, 

4  +  4+.  .  .  +4-0.  (7) 

Also,  since  equations  (6)  and  (7)  must  be  simultaneously 
satisfied  by  the  same  value  of  the  unknown,  being  the  most 
probable  value  in  either  case,  and  since  the  errors  4>  4»  •  •  • 
4  are  connected  only  by  the  relation  [J]  =  o,  we  necessari- 
ly have,  when  n  >  2* 


6  log  p(4)    _  6  log 


4  d4  4  d  Jt 

Clearing  of  fractions  and  integrating, 


—  .  .  .  —  k  suppose. 


*  When  «  =  2  it  reduces  to  an  identity. 


THE  LAW   OF    ERROR.  4! 

where  e  is  the  base  of  the  Napierian  system  of  logarithms 
and  c  is  a  constant. 

72    i 

Now,  since  </>  is  to  be  a  maximum,  —  ^  must  be  negative. 
But  when  <p  is  a  maximum  subject  to  the  condition  [J]  =  o, 

72    / 

then  —¥;  =  nk$.      Hence,  since  </>  is  positive,  k  must  be 
negative,  and,  putting  it  equal  to  -    —,  we  have 

_  A» 

y(  J)  =  ce    ^ 

the  law  of  error  sought. 

16.  In  this  expression  there  are  two  symbols  undeter- 
mined, c  and  IJL.  To  find  c.  Since  it  is  certain  that  all  of 
the  errors  lie  between  the  maximum  errors  -{-a  and  —a, 
we  have 


/+ 
a 


dA  = 


But  as  the  values  of  a  are  different  for  different  kinds  of 
observations,  and  as  we  cannot  in  general  assign  these  val- 
ues definitely,  we  must  take  -f-  °°  and  —  =o  as  the  extreme 
limits  of  error,  so  that  c  is  found  from 


/•+ 
c  I 

J       -00 


and  hence  (see  Art.  6) 


H  V2K 

and  the  law  of  error  may  be  written 

i 


ft  \2 


or  by  putting  //"  —  — 
* 


42  THE  ADJUSTMENT  OF  OBSERVATIONS. 

When  this  latter  form  is  used  it  is  only  for  greater  conveni- 
ence in  writing,  and  h  is  to  be  looked  on  as  a  mere  sym- 
bol standing  for 

W 

A" 

As  regards  //",  it  is  evident  that  for  e~  2^  to  be  a  possible 

J" 
quantity  —   must  be  an   abstract   number.     Hence  a  is  a 

I? 
quantity  expressed  in  the  same  unit  of  measure  as  A. 

Also,  from  the  form  of  the  function  <f>(^},  it  is  evident 
that  the  probability  of  an  error  J  will  be  the  larger  the 
larger  JJL  is,  and  vice  versa.  Hence  //  is  a  test  of  the 
quality  of  observations  of  different  series,  the  unit  being 
the  same. 

Again,  the  total  number  of  errors  in  a  series  being  n,  the 
number  between  J  and  A-\-dA  will,  from  the  definition  of 
probability,  be  n  tp(d)  dA.  Hence  the  sum  of  the  squares  of 
the  errors  A  in  the  same  interval  will  be  equal  to  nJ1  y(A)  dd, 
and  the  sum  of  the  squares  of  the  errors  between  the  limits 
of  error  -j-  a  and  —  a  will  be 


Extending  the  limits  of  error  ±  a  to  ±  °o  t  this  expression 
becomes,  after  substituting  for  <p(A)  its  value, 


/"•+0 

I 

J  ~« 


which  (see  Art.  6)  reduces  to  «//. 

Hence  //  is  the  mean  of  the  sum  of  the  squares  of  the 
errors  A  that  occur  in  the  series.  It  is  called  the  mean- 
square  error,  and  will  be  referred  to  as  the  m.  s.  e. 

17.  The  Principle  of  Least-Squares.  —  Having  de- 
fined the  symbols  c  and  //  in  the  expression  for  <p(A],  let  us 
return  to  Art.  15,  Eq.  5. 

If  the  observed  values  are  of  the  same  quality  through- 

_  LA.!3 
out,  p.  is  constant  and  the  product  <p  becomes  c*e    ^  .   This 


THE   LAW   OF  ERROR.  43 

product  is  evidently  a  maximum  when  [J11]  is  a  minimum  ; 
that  is,  if  we  assume  that  each  of  a  very  large  number  of  ob- 
served values  of  a  quantity  is  of  the  same  quality,  the  most  prob- 
able value  of  the  quantity  is  found  by  making  the  sum  of  the 
squares  of  the  errors  a  minimum. 

If  the  observed  values  are  not  of  the  same  quality,  (i  is 
different  for  the  different  observations,  and  the  most  prob- 
able value  of  the  unknown  would  be  found  from  the  maxi- 

-  f—  1 
mum  value  of  e    L2^J  .  that  is,  from  the  minimum  value  of 

pf-| 

I  -~a  J.     Thus  if  each  of  a  large  number  of  observed  values  of  a 

quantity  is  of  different  quality,  the  most  probable  value  of  the 
quantity  is  found  by  dividing  each  error  of  observation  by  its 
m.  s.  e.  and  making  the  sum  of  the  squares  of  the  quotients  a 
minimum. 

This  latter  includes  the  case  in  which  the  observed 
quantity  is  a  function  of  several  independent  unknowns 
whose  values  are  to  be  found.  For  if 


where  the  functions  /„/,,  .  .  .  are  of  the  same  form  and 
differ  only  in  the  constants  that  enter;  then  if  J,,  4,,  .  .  . 
denote  the  errors  of  observation,  we  have 

4  =/'(*.*  -  •  O-^i 

4  =/*(*,*  ...)-^ 

Hence  since  J,,  Ja,  .  .  .  are  functions  of  the  independent 
variables  x,  y,  .  .  .  we  must  have 


with  respect  to  the  variables  .r,  y,  .  .  .     But  the  differenti- 
ation of  this  equation  with  respect  to  xt  y,  .  .  .  and   the 
equating  of  the   differential    coefficients   to  zero,    gives   as 
7 


44  THE  ADJUSTMENT  OF  OBSERVATIONS. 

many  equations  as  unknowns,  from  which  equations  the 
most  probable  values  of  ,r,  y,  .  .  .  may  be  found. 

1 8.  Two  other  inferences  from  the   preceding   general 
principles  are  important : 

(a)  Since   in   a  series   of  observed    values   of  different 
quality  the  sum  of  the  squares  of  the  errors,  divided   by 
their  respective  m.  s.  e.  made  a  minimum,  leads  to  the  most 
probable  values,  it  follows  that  observed  values  of  different 
quality  are  put   on   a   common   basis   for   combination    by 
dividing  by  their  respective  m.  s.  e.     This  conclusion  will 
be  found  developed  in  Chapter  III. 

f  . 

(b)  Since  — r  is  an  abstract  number,  no  matter  what  the 

unit  of  measure  in  which  the  observed  values  are  expressed, 
it  follows  that  heterogeneous  measures  may  be  combined  in 
the  same  minimum  equation.  For  an  example  in  which  this 
is  fully  brought  out  see  Art.  169.  i 


The   Law   of  Error   of  a    Linear  Function  of  Independently 
Observed  Quantities. 

19.  We  have  found  the  law  of  error  in  the  case  of  a  quan- 
tity directly  observed,  and  which  may  be  a  function  of  one 
or  more  unknowns.  There  remains  the  question  as  to  the 
form  the  law  of  error  assumes  in  the  case  of  a  quantity,  F, 
which  is  a  linear  function  of  several  independently  observed 
quantities,  M»  M»  .  .  .  Mn;  that  is,  when 


where  alt  a  .....  are  constants. 

For  simplicity  in  writing  consider  two  observed  quanti- 
ties, M^  M^  only,  and  let  //„  /*2  be  their  m.  s.  e.  The  proba- 
bility of  the  simultaneous  occurrence  of  the  errors  J,  in  M^ 
and  4,  in  M,  is 


THE    LAW   OF   ERROR.  45 

Now,  an  error  4  in  Ml  and  an  error  Ja  in  M^  produce  an 
error  J  in  F,  according-  to  the  relation 

J  =  *,4+«,4  (2) 

and  this  relation  can  always  be  satisfied  by  combining  any 
value  of  4,  with  all  values  of  J,  ranging  from  —  oo  to  -f-co. 
The  probability,  therefore,  of  an  error  J  in  F  may  be  written 

<p(J)  dJ  =  HA  <Ut  f  'Y*'*'1-  W<W, 

7T  *s    -w 

But  from  (2),  and  since  J2  is  independent  of  J,, 

af  J  =  «„  ^Js 
Hence 


=h^d_J  /—  ^-^v-vC-^OV 

7T      <72  ^/    -w 

/,  /,  //        w         >-+=o    v^ 
_'tA«Je--*w+was*  I     i 

»   A  ./  - 


which  is  of  the  form 

h 


That  is,  the  law  of  error  of  the  function  F  is  the  same  as  that 
of  the  independently  measured  quantities  J/,,  J/2. 
The  m.  s.  e.  of  the  function  /MS  found  from 


that  is,  from 


Tliis  theorem  is  one  of  the  most  important  in  the  method  of 
least  squares,  and  will  be  often  referred  to. 


46  THE  ADJUSTMENT  OF    OBSERVATIONS. 

Ex.  To  find  the  m.  s.  e.  of  the  arithmetic  mean  of  n  equally  well  observed 
values  of  a  quantity  : 
We  have 

.  .  .  +Afn) 


Let  //„  =  m.  s.  e.  of  the  arithmetic  mean 

>u  =:  m.  s.  e.  of  each  observed  value 

Then  r 

/*«2  =  -5  (ju°-  +  /<5  +   ...  to  n  terms) 


That  is,  the  m.  s.  e.  of  the  arithmetic  mean  of  n  observations  is  —  =  part  of  that 

Vn 
of  a  single  observation. 


On   the   Comparison   of  the  Accuracy  of  Different   Series  of 

Observations. 

20.  The  Mean-Square  Error.— We  have  seen  in  Art. 
1 6  that  the  m.  s.  e.  //  affords  a  test  of  the  relative  accuracy 
of  different  series  of  observations.  This  test  was  suggested 
by  the  fundamental  formula  of  the  law  of  error,  and  is 
naturally  the  first  that  would  be  taken  for  that  purpose. 

The  value  of//  is  the  mean  of  the  sum  of  the  squares  of 
the  errors  in  a  series  between  the  extreme  limits  of  error, 
and  since  the  probability  of  an  error  is  the  number  of  cases 
favorable  to  its  occurrence  divided  by  the  total  number  of 
cases,  //  is  given  by  the  expression 


where  -f-  <*  and  —  a  are  the  limits  of  error. 

Hence  if  the  number  of  errors  n  is  a  very  large  number 
a  close  approximation  to  the  value  of  //  will  be  given  by 

A  "       A  z  A  a 

,t*  —  _L_  _L     2    _|_  -I-  — 

7      -   n         n  n 


The  difference  in  precision  of  these  two  values  of  //  will  be 
pointed  out  later.     (See  Art.  23.) 


THE   LAW   OF   ERROR.  47 

21.  The  Probable  Error. — A  second  method  of  deter- 
mining the  relative  precision  of  different  series  of  observa- 
tions is  by  comparing  errors  which  occupy  the  same  rela- 
tive position  in  the  different  series  when  the  errors  are 
arranged  in  order  of  magnitude.  The  errors  which  occupy 
the  middle  places  in  each  series  are,  for  greater  conveni- 
ence, the  ones  chosen. 

Let  the  errors  in  a  series,  arranged  in  order  of  magni- 
tude, be 

±  2a,  .  .  .   ±  r,  .  .   .  ot 

each  error  being  written  as  many  times  as  it  occurs ;  then 
we  give  to  that  error  r  which  occupies  the  middle  place, 
and  which  has  as  many  errors  numerically  greater  than  it 
as  it  has  errors  less  than  it,  the  name  of  probable  error.  If, 
therefore,  n  is  the  total  number  of  errors,  the  number  lying 

n 
between  -\-  r  and  —  r  is  — ,  and   the   number  outside    these 

limits  is  also  -.     In  other  words,  the   probability  that  the 

Z 

error  of  a  single  observation  in  any  system  will  fall  between 
the  limits  -\-r  and  —  r  is  -,  and  the  probability  that  it  will 

fall  outside  these  limits  is  also  — .     We  have,  therefore, 


V7r*S  -r 

from  which  to  find  r. 

If  we  put  // J  =  /,  and   the   value  t  =  p  corresponds  to 
J  =  r,  then 

2 


1  - 

Expanding  the  integral  in  a  series  as  in  (c)  Art.  6,  we  shall 
find  that  approximately  the  resulting  equation  is  satisfied  by 

p  =  0.47694 


48  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Now,  since 

hr  =  p  =  0.47694  and  /ifj.  \/2  —  z 

it  follows  that 

r  =  0.67457* 

=  -  n  roughly. 

0 

Hence  to  find  the  probable  error  we  compute  first  the 
mean-square  error  and  multiply  it  by  0.6745. 

As  a  check,  the  error  which  occupies  the  middle  place  in 
the  series  of  errors  arranged  in  order  of  magnitude  may  be 
found.  It  will  be  nearly  equal  to  the  computed  value,  if 
the  series  is  of  moderate  length. 

It  is  to  be  clearly  understood  that  the  term  probable  error 
does  not  mean  that  that  error  is  more  probable  than  any 
other,  but  only  that  in  a  future  observation  the  probability 
of  committing  an  error  greater  than  the  probable  error  is 
equal  to  the  probability  of  committing  an  error  less  than 
the  probable  error.  Indeed,  of  any  single  error  the  most 
probable  is  zero.  Thus  the  probability  of  the  error  zero  is 
to  that  of  the  probable  error  r  as 

•  g~ l 

Vx      Vn 
or 

T    •    ,,-(0'47694)2 

i  .  e 
or 

i  :  0.8 

The  idea  of  probable  error  is  due  to  Bessel  (Berlin.  Astron. 
Jahrb.,  1818).  The  name  is  not  a  good  one,  on  account  of 
the  word  probable  being  used  in  a  sense  altogether  different 
from  its  ordinary  signification.  It  would  be  better  to  use 
the  term  critical  error,  for  example,  as  suggested  by  De 
Morgan,  or  median  error,  as  proposed  by  Cournot. 

22.  The  Average  Error. — It  naturally  occurs,  as  a 
third  test  of  the  accuracy  of  different  series  of  observations, 
to  take  the  mean  of  all  the  positive  errors  and  .the  mean  of 
all  the  negative  errors,  and  then,  since  in  a  large  number  of 


THE  LAW   OF  ERROR.  49 

observations  there  will  be  nearly  the  same  number  of  each 
kind,  to  take  the  mean  of  the  two  results  without  regard  to 
sign.  This  gives  what  may  be  termed  the  average  error, 
It  is  usually  denoted  by  the  Greek  letter  rt. 

Reasoning  as  in  Art.  20,  we  have  approximately 

\A 

y  =  — 

n 

where  [J  is  the  arithmetic  sum  of  the  errors. 

An  expression  for  ^  in  terms  of  the  mean-square  error  // 
may  be  found  as  follows.  The  number  of  errors  between  J 
and  J  -f-  dJ  is 


and  the  sum  of  the  positive  errors  in  the  series  is 

n  C  A  <f(J)  dJ 

*s     0 

The  sum  of  the  negative  errors  being  the  same,  the  sum  of 
all  the  errors  is 

/+00 
A  (p(A)  dA 
) 

Hence 


2/1      f+°o 

—  ~7=  /          df 
VnJ  o 

fe-'Vl 


"  h 

the  relation  required. 

The  average  error  may,  as  stated  above,  be  directly  used 
as  a  test  of  the  relative  accuracy  of  different  series  of  ob- 
servations. Indeed,  I  think  it  should  be  more  used  tor  this 
purpose  than  it  is.  The  general  custom  is,  however,  to 
employ  it  as  a  stepping-stone  to  find  the  mean-square  and 
probable  errors.  This  can  be  done,  for  the  reason  that  it  is 
more  easy  to  compute  [J  than  [J-]. 


T1.  C 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


The  formulas  for  //  and  r  computed  in  this  way  are  as 
follows.     From  the  last  equation  preceding 


and  from  Art.  21 


=1.2533- 

n 


r  =  0.6745  n 
=  0.8453^ 


The  relations  connecting  /JL,  r,  and  rj  are  easily  remembered 
in  the  following  form  : 


These  relations  may  also  be  conveniently  arranged  in  tabu- 
lar form  : 


• 

t* 

r 

? 

f*  = 

I.OOOO 

1.4826 

1-2533 

r  = 

0.6745 

I.OOOO 

0.8453 

7j  = 

0.7979 

1.1829 

I.OOOO 

23.  We  have  seen  that  the  m.  s.  e.  p.  may  be  computed 
from  the  sum  of  the  squares  of  the  errors  and  also  from  the 
sum  of  the  errors  without  regard  to  sign.  In  the  deriva- 
tion of  each  formula  certain  approximations  have  been 
made.  The  question  then  arises  which  of  the  two  methods 
will  give  the  more  reliable  result.  This  will  be  shown  by 
the  ranges  in  the  two  determinations  of  the  value  of  p. 

(a)  We  proceed  to  find  first  the  m.  s.  e.  in  the  determi- 
nation of  //*  by  taking  the  approximate  formula 


--  (=  /V  suppose) 

ft 


THE   LAW   OF  ERROR.  51 

instead  of  the  rigid  formula 


If  we  put 

l  =  ri-f 

then,  since  //  is  the  exact  value,  I  will  be  the  error  of  // 
computed    from   the   errors   4>  J3  .    .   .  according   to   the 

formula  i  —  -.     Squaring,  we  have 
n 


Now,  letting  the   errors  J  assume  all  possible  values,  the 
average  value  of  the  fourth  powers  is  (see  Art.  6) 


The  number   of  the   products  4*4*1  4*4'»  •  •  •   being  the 
number  of  combinations  of  n  things,  two  at  a  time,  is  -'— — 
and  the  average  of  the  values  is 

n(n-i)  \    2h 
2 

that  is  (Art.  20), 

"„'  P 


j   21     r 

\    Vx  J  o 


- 


The  average  value  of  ft  LI  is  //*.     Hence  finally 


n         n*        2 


52  THE  ADJUSTMENT   OF   OBSERVATIONS. 

and 


or 

*•=*(' 

=  /jLii  ±  —  —  )   when  n  is  very  large. 

V  V2H' 

(b)  In  the  second  place,  for  the  average  error  37  we  pro- 
ceed in  precisely  the  same  way.     We  have 


Let 
Then 


=  /£.  4.     ^^  - 

'" 


n    '   wa         2  TT 


which  gives  the  error  in  37. 
Also,  since 


the  error  in  //  is 

a       /7T  .  /        2    iU  ,  .         .A  — 2 
~=  A/  -  \f   l  —  -I  that  is,  //  A/ 

I/TJJ     ^      2     K  7T  f          2« 

Hence  by  this  method   of   computing  JJL  the  value  of  //  is 
contained  between  the  mean  limits 


THE    LAW   OF   ERROR.  53 

No\v,  since  r  —  2  >  i,  the  limits  in  the  latter  case  are  the 
larger,  and  we  therefore  conclude  that  the  former  method 
of  computing  /*  is  the  better  of  the  two. 
24.   From  the  equation 


we  may  derive  a  test  of  the  validity  of  the  law  of  error,  and 
a  rather  curious  one.  For  /*  and  y  may  be  determined 
from  measurements,  and  if  the  experimental  values  found 
satisfy  the  equation 

£-* 

r/        2 

we  must  conclude  that  the  theory  is  correct.  This  may  be 
classed  as  an  additional  a  posteriori  proof  to  that  given  in 
Art.  51. 

25.  Whether  we  should  use  the  m.  s.  e.  or  the  p.  e.  in 
stating  the  precision  is  largely  a  matter  of  taste.  Gauss 
says  :  "  The  so-called  probable  error,  since  it  depends  on 
hypothesis,  I,  for  my  part,  would  like  to  see  altogether 
banished  ;  it  may,  however,  be  computed  from  the  mean  by 
multiplying  by  0.6744897."  On  the  other  hand,  the  Inter- 
national Committee  of  Weights  and  Measures  decided  in 
favor  of  the  probable  error:  "  It  has  been  thought  best  in 
this  work  that  the  measure  of  precision  of  the  values  ob- 
tained should  always  be  referred  to  the  probable  error  com- 
puted from  Gauss'  formula,  and  not  to  the  mean  error." 
(Proch  Verbaux,  1879,  P-  77-) 

In  the  United  States,  in  the  Naval  Observatory,  the 
Coast  Survey,  the  Engineer  Corps,  and  the  principal  ob- 
servatories, the  p.  e.  is  used  altogether.  So,  too,  in  Great 
Britain,  in  the  Greenwich  Observatory,  the  Ordnance  Sur- 
vey, etc.  In  the  G.  T.  Survey  of  India  the  m.  s.  e.  is  used, 
for  the  reason  given  by  Gauss  above.  Among  German 
geodelicians  and  astronomers  the  m.  s.  e.  is  very  generally 
employed. 

In  this  book  the  m.  s.  e.  will  be  used  in  the  text,  and  the 
m.  s.  e.  and  p.  e.  in  the  examples  indifferently. 


54 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


The  Probability  Curve. 

26.  The  principles  laid  down  in  the  preceding  articles 
may  be  illustrated  geometrically  as  follows: 

We  have  seen  that  in  a  series  of  observations  the  proba- 
bility that  an  error  will  lie  between  the  values  x  and  x-\-dx 
is  given  by  the  expression 


h 


dx 


Fig.l 


Now,  if  O  is  the  origin  of  co-ordinates,  and  a  series  of  errors, 

x,  are  represented  by  the  distances 
from  O  along  the  axis  of  abscissas 
OX,  positive  errors  being  taken  to 
the  right  of  O  and  negative  errors 
to  the  left,  then  the  probability,  in 
a  future  observation,  of  an  error 
falling  between  x  and  x  -f-  dx  will 

be  represented  by  the  rectangle  whose  height  is  — ^  e~h<t^ 

Vn 

and  width  dx,  or,  more  strictly,  by  the  ratio  of  this  rectangle 
to  the  sum  of  all  such  rectangles  between  the  extreme 
limits  of  error.  This  sum  we  have  for  convenience  already 
denoted  by  unity. 

Hence  for  a  series  of  observations  whose  quality  is 
known,  by  giving  to  x  all  values  from  -j-oo  to  —  oo  and  draw- 
ing the  corresponding  ordinates,  we  shall  have  a  continuous 
curve  whose  equation  may  be  written 

fi  .  _£2~2 


This  curve  is  called  the  probability  curve. 

27.  To  Trace  the  Form  of  the  Curve. — Since  x 
enters  to  the  second  power,  and  y  to  the  first  power,  the 
curve  is  symmetrical  with  respect  to  the  axis  of  y,  and  the 
form  of  the  equation  shows  that  it  lies  altogether  on  one 


THE    LAW    OK    ERROR.  55 

side  of  the  axis  of  x.     Also,  when  x  =  o,  -  -=o;  that  is,  the 

ax 

tangent  at  the  vertex  is  parallel  to  the  axis  of  x. 

As  x  increases    from  o  the  values  of  y  continually  de- 
crease.    When  x  =  ±  co  ,  then 

j  Jy 

y  =  o  and  -f-  —  o 
ax 

showing  that  the  axis  of  x  is  an  asymptote. 
Again,  since 


-*        -—  -- 

dx        t/jj. 

there  is  a  point  of  inflection  when 

i 

X  —   -:—-=.  =  fJ. 

h^2 
and  the  m.  s.  e.  is  therefore  the  abscissa  of  the  point  of  in- 


flection.    Also,  when  x  =  o,  —^  is  negative,  showing  that  the 

dx* 

ordinate  at  the  vertex  is  the  maximum  ordinate.  Hence 
the  curve  is  of  the  form  indicated  in  Fig.  i,  OA  represent- 
ing the  maximum  ordinate  and  CU/the  m.  s.  e. 

The  values  of  //,  that  is,  of  -    p.,  being  different  for  diflfer- 

l*V  2 

ent  series  of  observations,  the  form  of  the  curve  will  change 
for  each  series,  and  the  curve  may  be  plotted  to  scale  from 
values  of  y  corresponding  to  assumed  values  of  .r. 

In   plotting   the   curve,  since   the  maximum   ordinate  at 

the  vertex  -    -  enters  as  a  factor  into  the  values  of  each  of 

\'  - 

the  other  ordmates,  its  value  may  be  arbitrarily  assumed. 
We  may  therefore  adopt  a  scale  for  plotting  the  ordinates 
different  from  the  scale  by  which  the  abscissas  are  plotted, 
in  order  to  show  the  curve  more  clearly. 

The  form   of  the  curve  is  in  accordance  with  the  prin- 
ciples already  laid  down  in  deducing  the  law  of  error,  and 


56  THE  ADJUSTMENT   OF  OBSERVATIONS. 

could  have  been  derived  from  them  directly.  Thus  that 
small  errors  are  more  probable  than  large  is  indicated  by 
the  element  rectangle  areas  being  greater  for  values  of  x 
near  zero  than  for  values  more  distant ;  that  very  large 
errors  have  a  very  small  probability  is  indicated  by  the 
asymptotic  form  of  the  curve,  and  that  positive  and  negative 
errors  are  equally  probable  is  indicated  by  its  symmetrical 
form  with  respect  to  the  axis  of  y. 

28.  The  area  of  the  curve  of  probability  being  the  sum 

of  the  rectangles  —j=.e"kj^dx,  for  values   of  x  extending. 

V  7T 

from  -j-  oo  to  —  oo  we  have  denoted  by  unity.  If,  then,  we 
represent  by  the  area  of  this  curve  the  total  number  of 
errors  that  occur  in  a  series  of  observations,  it  follows  from 
the  definition  of  probability  that  the  area  included  between 
certain  assigned  limits  would  represent  the  number  of 
errors  to  be  expected  in  the  series  between  the  values  of 
those  limits. 

Thus  if  O  is  the  origin,  the  area 
to  the  right  of  OA  would  represent 
the  number  of  positive  errors  and 
the  area  to  the  left  of  OA  the  num- 
ber of  negative  errors.  The  area 

OPP'A  would  represent  the  number 

0    P  R  X 

of  positive  errors  less  than   OP,  the 

area  PRR'P  the  number  that  lie  between  OP  and  OR. 

If  the  area  AOPP'  is  equal  to  one-half  the  total  area 
AOX.  then  the  number  of  positive  errors  less  than  OP 
would  be  equal  to  the  number  greater  than  OP.  Hence' 
OP  would  represent  the  probable  error.  If  OQ  be  taken 
equal  to  OP,  the  area  PQQ ' P'  would  represent  the  number 
of  errors  numerically  less  than  the  probable  error. 

29.  The  average  error  57  may  be  illustrated  as  follows: 
If  x^  is  the  abscissa  of  the  centre  of  gravity  of  any  curve 
referred  to  the  axes  of  Jf  and  Y,  then 

_  fyx  dx 


THE   LAW   OF   ERROR.  57 

Applying  this  to  the  curve  of  probability  and  calling  the 
whole  area  unity,  we  have 


h  \/~ 
=  y,  the  average  error. 

The   Law   of  Error  applied  to  an  Actual  Series  of  Observa- 
tions. 

We  here  bridge  over  the  gulf  between  the  ideal  series 
from  which  we  have  derived  the  law  of  error  and  the  actual 
series  with  which  we  have  to  deal  in  practical  work,  and 
which  can  only  be  expected  to  come  partially  within  the 
range  of  the  law  constructed  for  the  ideal. 

30.  Effect  of  Extending  the  Limits  of  Error  to 

±00. — The  expression— -=e~h*^dA  gives   the  value  of  the 

\  Tt 

probability  of  an  error  between  J  and  d-\-dJ  in  an  ideal 
series  of  observations  where  the  values  are  continuous  be- 
tween limits  infinitely  great.  In  all  actual  series  the  possible 
error  is  included  within  certain  finite  limits,  and  the  proba- 
bility of  the  occurrence  of  an  error  beyond  those  limits  is 
zero.  Practically,  however,  the  extension  of  the  limits  of 
error  to  ±  °°  can  make  no  appreciable  difference  in  either 
case,  as  the  function  <p(J)  decreases  so  rapidly  that  we  can 
regard  it  as  infinitesimal  for  large  values  of  J ;  in  other 
words,  the  greater  number  of  errors  is  in  the  neighborhood 
of  zero,  and  therefore  the  most  important  part  is  the  part 
covered  by  both.  This  has  been  illustrated  geometrically 
in  the  discussion  of  the  probability  curve,  and  will  now  be 
developed  from  another  point  of  view. 


58  THE   ADJUSTMENT   OF   OBSERVATIONS. 

We   have   for  the  probability   of  the  occurrence  of  an 
error  not  greater  than  a,  in  a  series  of  observations, 

2h 


This  may  be  put  in  the  form  (/  = 

2 


and  is  usually  denoted  by  the  symbol  6(t). 

The  method  of  evaluating-  S(f)  has  been  explained  in 
(c)  Art.  6.  In  Table  I.  will  be  found  the  values  of  the  func- 
tion S(t)  corresponding  to  the  argument  — ,  that  is,  -.  The 

P  r 

reason  for  arranging  the  table  in  this  way  is  that  it  is  more 

convenient  to  compute  —  than  —  p. 

r  r 

The  probability  that  an  error  exceeds  a  certain  error  a 
is  i  —  #(/),  and  may  be  found  from  Table  I.  by  deducting 
the  tabular  value  from  unity.  Thus  we  have  the  probability 
that  a  is  greater  than  r  is  0.5,  than  2r  is  0.1773,  than  y  is 
0.0430,  than  4r  is  0.0070,  than  ^r  is  0.0007,  than  6r  is  o.oooi. 

Hence  in  10,000  observations  we  should  expect  only  one 
error  greater  than  6r,  in  1,000  only  one  greater  than  $r,  in 
100  only  one  greater  than  4?%  and  in  25  only  one  greater 
than  y.  If  in  any  set  of  observations  we  found  results 
much  at  variance  with  these  we  could  assume  that  they 
arose  from  some  unusual  cause,  and  should,  therefore,  be 
specially  examined.  As  in  practice  the  number  of  observa- 
tions in  any  case  is  usually  under  100,  we  are  eminently  safe 
in  taking  the  maximum  error  at  about  $r  or  3//. 

We  are  now  in  a  position  to  estimate  the  change  intro- 
duced by  replacing  c  as  found  from 


THE  LAW   OF   ERROR.  59 


h 
by  —:=  as  found  from 


in  the  investigation  of  the  law  of  error. 
These  equations  may  be  written 


Hence 

Jl      1  2/1        /*<*>  \ 

c=  -    -.(1+    —=.    I      e-A**3  tU  I  approximately. 

\  -     \  \   -  J   a  I 

and  taking  a  =  $r,  we  have,  from  Table  I., 

//     , 
c"~^  (2-°-9993) 

h     • 
=  1.001  - 

T  7T 

Hence  the  difference  being  less  than   ~T—  :  of  the  quantity 

I  OOO 

sought,  the  approximate  value  of  c  found  by  extending  the 
limits  ±  a  to  ±  oo  may  be  considered  satisfactory. 

31.  Various  Laws  of  Error.  —  We  have  taken  the 
arithmetic  mean  of  a  series  of  observed  values  of  a  quantity 
made  under  like  conditions  as  the  most  plausible  value  of 
the  quantity.  The  supposition  of  each  observed  quantity 
being  subject  to  the  same  law  of  error  leads  to  the  mean  as 
the  most  probable  value.  "  The  method  of  least  squares  is, 
in  fact,  a  method  of  means,  but  with  some  peculiar  char- 
acters. The  method  proceeds  upon  this  supposition,  that 
all  errors  are  not  equally  probable,  but  that  small  errors  are 
more  probable  than  large  ones."  * 

Now,  in  an  ordinary  series  we  assume  a  good  deal  when 
we  take  each  observation  of  the  series  as  subject  to  the  same 

*  Whewell,    History  v/the  Induclirc  Sciences,  vol.  ii. 


60  THE  ADJUSTMENT  OF   OBSERVATIONS. 

special  law  of  error  —  the  exponential  law.  We  can  certainly 
conceive  of  laws  different  from  this  one.  It  is  more  probable 
that  each  set  of  observations  has  its  own  law  depending  on 
instrument,  observer,  and  conditions.  If  we  could  go  back 
to  the  sources  of  error  we  could  find  this  law  in  each  case. 
Let  us  follow  out  this  idea  in  a  few  simple  cases  and  see  to 
what  it  leads  : 

32.  First  take  the  case  where  all  errors  are  equally  prob- 
able —  that  is,  where  J  can  with  the  same  probability  as- 
sume all  values  between  -f-  a  and  —  #,  the  extreme  limits  of 
error. 

Since  a  is  the  maximum  error, 

s*  +  a 

(p(A)  I      dd=i,  <p(A)  being  constant, 

»/    -a 

and  therefore 

K^)=— 

20. 

the  law  of  error. 

For  the  m.  s.  e.  //  we  have 


r* 
=  / 

J    -a 


3 
Also 


/* 
a. 


2a 


2 

The  p.  e.  is  found  from  the  relation 


or 


r+rdA     i 

/ 

J    -r     2a          2 


THE   LAW    OF   ERROR. 


6l 


and 


Fig.3 


that  is,  the  p.  e.  is  half  the  max.  error. 

This  is  also  evident  geometrically,  for  the  curve  of  error 

y  =•  constant 

is  a  straight  line  parallel  to  the 
axis  of  abscissas. 

Hence   by   definition    we   find 
the  p.  e.,  by  bisecting  the  area,  to 


be  -.     The  p.  e.  would  be  repre- 

o       p  x 

sented  in  the  figure  by  OP. 

33.  Next  consider  an  error  to  arise  from  two  independent 
sources,  x,  y,  each  of  which  can  with  the  same  probability 
assume  all  values  between  -f-  a  and  —  a,  so  that  for  the  total 
error  J  we  have 


To  find  <f(J),  the  law  of  error:  If  J  and  J  -f-  dJ  denote  two 
consecutive  errors,  then  since  x  and  y  have  values  between 
-\-a  and  —a  with  interval  dJ,  each  of  the  quantities  x,  y  has 

'—  possible  values,  and  the  whole  number  of  possible  causes 


The  causes  which  are  favor- 


r  .     2a         2a         4<i 

oferrorisZ7xZ(or(ZO" 

able  for  an  error  of  the  magnitude  -f-  -/  answer  to  all  possible 
values  of  x,  y  which  together  give  -\-  J ;  namely, 

for  x,  J  —  a,   J  —  a  -f-  dJ,  A  —  a  -(-  2dJ,  ...          a  —  dJ,         a 
for  y,          a,  a—  dJ,  a  —  2dJ,  .  .  .  J  —  a  -\-  dJ,  J  —  a 

and  whose  number  is  therefore — . 

In  the  same  way  we  find  the  number  of  causes  favorable 
for  the  error  of  the  magnitude  —  J  to  be  ~-—^- — 


62 


THE  ADJUSTMENT  OF   OBSERVATIONS. 


Hence  the  probability  <p(A)  dd  of  an   error   between 
and  J  -f-  */J  is  given  by 


that  is, 


=  —  it  —  when  J  lies  between  —  2«  and  o 
4a 

O//  _    A 

=  -  —  -  —  when  J  lies  between  o  and  -{-  2a. 


For  the  m.  s.  e.  we  have 


r  3.  »^         r>  f  ~  +  J 

J  o          4«2  '  J  _  2a         4« 


For  the  p.  e., 

fr2 
/ 
«/  o 


s*o  2a  -\-  A 
/       -~ 
»/  -  r     4» 


Geometrically,  the  equations 

J  —  2d!  —  X 

y  =  2a  -f-  x 


Fig.4 


represent  two   straight   lines 
which  cut  the  axes  of  x  and  y 
at  an  angle  of  45°,  and  there- 
fore the  curve  of  probability 
is  as  in  the  figure.      If  in  the 
figure  OA  —  2a,  then  OP  rep- 
resents the  p.  e. 
34.  From    the   preceding  we    may  derive  an   important 
practical  point.     In  an  ordinary  seven-place  log.  table  the 
seventh  place  is  never  in  error  by  more  than  o  5.     Hence, 


0     P 


THE   LAW    OF   ERROR.  63 

this  being  the  maximum  error,  the  p.  e.  of  a  log.  as  given  in 
the  tables  is  0.25  in  the  seventh  place.  The  interpolated 
value  at  the  greatest  distance  from  a  tabular  value  is  the 
mean  between  two  tabular  values.  Its  p.  e.,  from  Art.  33,  is 

(2  —    1/2)  X  0.25  =0.15 

Hence  the  p.  e.  of  the  log.  of  a  number  may  be  taken  0.2  in 
the  seventh  decimal  place.  The  p.  e.  of  the  number  corre- 
sponding to  this  log.  is  (Ex.  i,  Art.  7) 

= approx. 

io8  X  mod.       22  X  10 

Suppose  now  that  we  are  computing  a  chain  of  triangles 
starting  from  a  measured  base.  The  p.  e.  of  the  base  may 

be  taken  as of  its  length.     Hence  the  error  arising 

1,000,000 

from  this  source  is  22  times  that  to  be  expected  from  the 
log.  tables.  Again,  the  triangulation  will  be  most  exact,  and 
therefore  the  test  most  severe,  when  the  angles  of  each  tri- 
angle are  equal  to  60°.  Now,  the  change  in  log  sin  60° 
corresponding  to  a  change  of  i/ris  12.2  in  units  of  the  seventh 
decimal  place.  And  in  a  primary  triangulation  an  angle 
may,  with  the  instruments  now  in  use,  be  measured  with  a 
p.  e.  of  o".25-  Hence 

p.  e.  of  log  sin  60=  12.2  X  0.25 

=  3.0  in  units  of  the  seventh  decimal  place, 

which  p.  e.  is  15  times  greater  than  that,  arising  from  the 
log.  tables. 

For  the  solution  of  triangles,  therefore,  we  conclude  that, 
with  our  present  means  of  measurement,  seven-place  tables 
are  sufficient.  The  common  practice  is  to  carry  out  to 
eight  places  to  give  greater  accuracv  in  the  seventh  place, 
and  then  drop  the  eighth  place  in  stating  the  final  result. 
(See  Struve,  Arc  du  Mcridicn,  vol.  i.  p.  94.) 

35.  If  an  error  J  arises  from  three  independent  sources 
of  the  same  kind,  each  of  which  can  with  the  same  proba- 
bility assume  all  values  between  +  a  and  --  a,  then,  the 


64  THE   ADJUSTMENT   OF   OBSERVATIONS. 

maximum   error  being  3*?,  we  have,  from  similar  reasoning 
to  that  employed  in  Art.  33, 

<p(A)  =  -^  —     '  '  when  J  is  between  -f-  3«  and  -J-  a 
lua 

2  (2 

^(  J)  =  -3_  I—  when  J  is  between  -f-  a  and  —  # 
8rt 

=  ^=  —  7r*~  wnen  ^  ^  between  —  a  and  —  3# 
Also 


=  2  A 

Jo 


and 


/; 


Sl>* 


The  curve  of  probability  consists  of  three  parts,  as  in 
the  figure : 

OA  =  OB  —  30  OC=OD—a 
There  are  common  tangents  to 
the  two  branches  at  E  and  F,  and 
the  curve  touches  the  axis  of  X 

,  at  A  and  B.     The  p.  e.  is  repre- 

0    PC      \        A 

sented  by  OP. 

36.  A  consideration  of  the  results  obtained  in  Arts.  31-35 
will  show  that  the  more  numerous  the  sources  of  error  as- 
sumed the  nearer  we  approach  the  results  obtained  from 
the  Gaussian  law  of  error.  Thus  for 

r  T 

one  source       —  1=  0.87        -  =  i.oo 

y  T 

two  sources     -  =  0.72       -  =  0.88 

T  y 

three  sources  -  =  0.71        —  =  0.87 

Gaussian  law  —  =  0.67         •  =10.85 


THE   LAW   OF   ERROR. 


The  forms  of  the  curves  of  probability  show  the  same 
approach  to  coincidence.  Starting  with  a  straight  line  as 
the  curve  for  a  single  source  of  error,  we  approach  quite 
closely  to  the  Gaussian  probability  curve,  even  with  so 
small  a  number  of  sources  of  error  as  three.  Hence  we 
should  expect  that,  starting  from  the  postulate  of  an  error 
being  derived  from  the  combined  influence  of  a  very  large 
number  of  independent  sources  of  error,  we  should  arrive  at 
the  Gaussian  law  of  error.  A  complete  demonstration  of 
this  by  Bessel,  to  whom  the  idea  itself  is  due,  will  be  found 
in  Astron.  Nachr.,  Nos.  358,  359.  The  elementary  proof 
given  in  Art.  33  for  the  simple  case  of  two  sources  of  error 
is  due  to  Zachariae. 

37.  Experimental  Proof  of  the  Law  of  Error.— 
The  same  point  was  brought  out  experimentally  in  a  series 
of  researches  by  Prof.  C.  S.  Peirce,  of  the  U.  S.  Coast  Sur- 
vey.* He  employed  a  young  man,  who  had  had  no  previous 
experience  whatever  in  observing,  to  answer  a  signal  consist- 
ing of  a  sharp  sound  like  a  rap,  the  answer  being  made  upon 
a  telegraph  operator's  key  nicely  adjusted.  Five  hundred 
observations  were  made  on  each  of  twenty-four  days.  The 
results  for  the  first  and  last  days  are  plotted  below.  In  the 

Fig.6 


0?35 


0?10 


0*30 


Fig.7 


0?20  0?25  0*30 

*  Coast  Surrey  Report,  1870,  appendix  21. 


66  THE  ADJUSTMENT   OF  OBSERVATIONS. 

figures  the  abscissas  represent  the  interval  of  time  between 
the  signal  and  the  answer,  the  ordinates  the  number  of  ob- 
servations. The  curve  is  a  mean  curve  for  every  day,  drawn 
by  eye  so  as  to  eliminate  irregularities  entirely.  After  the 
first  two  or  three  days  the  curve  differed  very  little  from 
that  derived  from  the  theory  of  least  squares.  On  the  first 
day,  when  the  observer  was  entirely  inexperienced,  the  ob- 
servations scattered  to  such  an  extent  that  the  curve  had  to 
be  drawn  on  a  different  scale  from  that  of  the  other  days. 
38.  General  Conclusions. — On  the  whole,  though  we 

cannot  say  that  the  formula  -—=  e~^^  will  truly  represent 

V  7T 

the  law  of  error  in  any  given  series  of  observations,  we  can 
say  that  it  is  a  close  approximation. 

When  in  a  series  of  observations  we  have  exhausted 
all  of  our  resources  in  finding  |;he  corrections,  and  have 
applied  them  to  the  measured  values,  the  residuum  of  error 
may  fairly  be  supposed  to  have  arisen  from  many  sources  ; 
and  we  conclude  from  the  foregoing  investigations  that,  of 
any  one  single  law,  the  best  to  which  we  can  consider  the 
residual  errors  subject,  and  the  best  to  be  applied  to  a  set  of 
observations  not  yet  made,  is  the  exponential  law  of  error. 

The  general  theorem  of  Art.  17  may  therefore  be  applied 
to  a  limited  series  and  be  written :  If  the  observed  values  of  a 
quantity  are  of  different  quality,  the  most  probable  value  is  found 
by  dividing  each  residual  error  by  the  m.  s.  e.  and  making 
the  sum  of  the  squares  of  the  quotients  a  minimum  ;  if  of  the 
same  quality,  the  most  probable  value  is  the  arithmetic  mean  of 
the  observed  values. 

If  a  set  of  observations  shows  a  marked  divergence  from 
this  law  a  rigid  examination  will  reveal  the  necessity,  in 
general,  of  applying  some  hitherto  unknown  correction. 
Thus  in  the  earlier  differential  comparisons  of  the  compen- 
sating base-apparatus  of  the  United  States  Lake  Survey  with 
the  standard  bar  packed  in  ice,  the  observed  differences  did 
not  follow  the  law  of  error,  as  it  was  fair  to  suppose  that 
they  should,  the  bars  being  compensating.  There  was  in- 


THE   LAW    OF   ERROR.  67 

stead  a  regular  daily  cycle :  some  one  source  of  error  so  far 
exceeded  the  others  that  it  overshadowed  them.  A  study 
of  the  results  was  made,  and  the  law  of  daily  change  dis- 
covered, which  gave  a  means  of  applying  a  further  correc- 
tion. The  work  done  later,  after  taking  account  of  this 
new  correction,  showed  nothing  unusual. 

• 
Classification  of  Observations. 

39.  For  purposes  of  reduction  observations  may  be  di- 
vided into  two  classes — those  which  are  independent,  being 
subject  to  no  conditions  except  those  fixed  by  the  observa- 
tions themselves,  and  those  which  are  subject  to  certain 
conditions  outside  of  the  observations,  as  well  as  to  the  con- 
ditions fixed  by  the  observations.  In  the  former  class,  be- 
fore the  observations  are  made,  any  one  assumed  set  of 
values  is  as  likely  as  any  other;  in  the  latter  no  set  of  values 
can  be  assumed  to  satisfy  approximately  the  observation 
equations  which  does  not  exactly  satisfy  the  a  priori  condi- 
tions. 

For  example,  suppose  that  at  a  station  O  the  angles 
A  OB,  AOC  are  measured.  If  the  measures  of  each  angle 
are  independent  of  those  of  the  other,  the  angles  are  found 
directly. 

The  angle  BOC  could  be  determined  from  the  relation 

AOC=AO£  +  BOC 

The  unknown  in  this  case  may  be  said  to  be  observed  in- 
directly, and  therefore  independent  observations  may  be 
classed  as  direct  and  indirect.  The  former  class  is  a  special 
class  of  the  latter. 

But  if  the  angle  BOC  is  observed  directly  as  well  as 
A  OB,  A  OC,  then  these  angles  are  no  longer  independent, 
but  are  subject  to  the  condition  that  when  adjusted 

AOC=  AOB  +  BOC 

andno  set  of  values  can  be  assumed  as  possible  which  does 
not  exactly  satisfy  this  condition. 

10 


68  THE  ADJUSTMENT   OF   OBSERVATIONS. 

The  observations  in  this  case  are  said  to  be  conditioned. 
Though  we  have,  therefore,  strictly  speaking,  only  two 
classes  of  observations,  we  shall,  for  simplicity,  divide  the 
first  into  two  and  consider  in  order  the  adjustment  of 

(1)  Direct  observations  of  one  unknown. 

(2)  Indirect    observations    of   several    independent   un- 
knowns. 

(3)  Condition  observations. 


CHAPTER   III. 

ON    THE    ADJUSTMENT    OF    DIRECT    OBSERVATIONS    OF    ONE 
UNKNOWN    QUANTITY. 

IN  the  application  of  the  ideal  formulas  of  Chapter  II.  to 
an  actual  series  of  observations  we  shall  begin  with  a  single 
quantity  which  has  been  directly  observed.  We  shall  con- 
sider two  cases  —  first,  when  all  of  the  observed  values  are 
of  equal  quality,  and,  next,  when  they  are  not  all  of  equal 
quality. 

A.    Observed  Values  of  Equal  Quality. 

40.  The  Most  Probable  Value  ;  the  Arithmetic 
Mean.  —  We  have  seen  that  in  a  series  of  directly  observed 
values  Mlt  M^  .  .  .  Mn  of  equal  quality  the  most  probable 
value  Fof  the  observed  quantity  is  found  by  taking  the 
arithmetic  mean  of  these  values;  that  is, 


It  has  also  been  shown  that  the  same  result  will  follow 
by  making  the  sum  of  the  squares  of  the  residual  errors  a 
minimum.  Thus  the  observations  give  the  equations, 


(2) 


and  V  is  to  be  found  from 

»,'  +  ».•+  .  .  .  +  *>»'=:  a  min.  (3) 

that  is,  from 

.  .  .  +(F-J/M)'  =  amin.   (4) 


70  THE   ADJUSTMENT   OF   OBSERVATIONS. 

By  differentiation  of  (4) 


o      (5) 

and  V—  (6) 

n 

In  practice  it  would  evidently  be  simpler  to  find  the  value 
of  the  unknown  by  taking  the  arithmetic  mean  of  the 
observed  values  directly  rather  than  to  form  the  obser- 
vation equations  and  find  it  by  making  the  sum  of  the 
squares  of  the  residuals  a  minimum. 

It  is  useful  to  notice,  for  purposes  of  checking,  that  Eq. 
(5)  may  be  written 

M=o  (7) 

41.  As  the  observed  values  M  are  often  numerically 
large  and  not  widely  different,  the  arithmetical  work  of 
finding  the  mean  may  be  shortened  as  follows: 

A  cursory  examination  of  the  observations  will  show 
about  what  the  mean  value  V  must  be.  Let  X'  denote  this 
approximate  value  of  F,  which  may  conveniently  be  taken 
some  round  number.  Subtract  X'  from  each  of  the  ob- 
served values  J/j,  Mt,  .  .  .  Mn  in  succession,  and  call  the 
differences  /t,  /„,...  /„  respectively.  Then 

Ml  —  Xt=l1 

M,-X'  =  ls  (i) 

jf.--r=4 

By  addition, 


n 
—  X'-{-.v'  suppose.       (2) 

Hence  all  that  we  have  to  do  is  to  take  the  mean  x'  of  the 
small  quantities  /„  /2,  .  .  .  /„,  and  add  the  assumed  value  X' 
to  the  result. 


DIRECT    OBSERVATIONS.  71 


Ex.  The  measured  values  of  an  angle  are 

177"      21'      5". 80 

177°     21'     7".35 

177°      2l'      4". 28 

find  the  mean. 

It  is  sufficient  to  find  the  mean  of  the  seconds  and  carry  in  the  degrees 
and  minutes  unchanged. 

42.  Control  of  the  Arithmetic  Mean. — In  least 
squares,  as  in  all  computations,  it  is  important  to  have  a 
check  or  control  of  the  numerical  work.  This  is  specially 
desirable  when  a  computation  takes  several  weeks,  or  it 
may  be  months,  to  complete  it.  In  long  computations  it  is 
better  for  two  computers  to  work  together,  using  different 
methods  whenever  possible,  and  to  compare  results  at  in- 
tervals. But  even  this  is  not  an  absolute  safeguard  against 
mistakes,  as  it  sometimes  happens  that  both  make  the  same 
slip,  as,  for  example,  writing  -f-  for  — ,  or  vice  versa.  Hence, 
even  if  the  computation  is  made  in  duplicate,  it  is  advisable 
to  carry  through  an  independent  check  which  may  be  re- 
ferred to  -occasionally.  In  computations  not  duplicated  a 
control  is  essential. 

A  control  of  the  accuracy  of  the  arithmetic  mean  of  a  set 
of  observed  values  of  the  same  quantity  is  afforded  by  the 
relation 

M=o 

that  is,   that  the  sum   of  the  positive  residuals  should  be 
equal  to  the  sum  of  the  negative  residuals. 

If,  however,  in  finding  the  arithmetic  mean,  the  sum  [J/] 
of  the  observed  quantities  was  not  exactly  divisible  by  their 
number  n,  the  sums  of  the  positive  and  negative  residuals 
would  not  be  equal,  but  the  amount  of  the  discrepancy 
could  easily  be  estimated  and  allowed  for.  For  if  the  value 
of  the  mean  taken  were  too  large  by  a  certain  amount,  the 
positive  residuals  would  each  be  too  large,  and  the  negative 
residuals  also  too  large,  by  that  amount.  Hence  the  dis- 
crepancy to  be  expected  would  be  n  times  the  amount  that 
the  approximate  quotient  taken  as  the  mean  differed  from 
the  exact  quotient. 


72  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  In  the  telegraphic  determination  of  the  difference  of  longitude 
between  St.  Paul  and  Duluth,  Minn.,  June  15,  1871,  the  following  were  the 
corrections  found  for  chronometer  Bond  No.  176  at  ish.  5im.  sidereal  time 
from  the  observations  of  21  time  stars.  (Report  Chief  of  Engineers  U.  S.  A.^ 
1871.) 


M 

V 

vv 

s. 

s. 

—  8.78 

+  0.04 

0.0016 

.76 

,   +   .02 

4 

.85 

+   .11 

121 

.78 

+  .04     s. 

16 

•  51 

—  O.23 

529 

.64 

—   .IO 

IOO 

.68 

—   .06 

36 

.63 

-   .11 

121 

•58 

-   .16 

256 

.80 

+   .06 

36 

•  75 

+   .01 

I 

.78 

+  .04 

16 

.96 

+   .22 

484 

.64 

-  .IO 

IOO 

•  65 

-  .09 

81 

•83 

+   .09 

81 

.70 

-   -04 

16 

.64 

—  O.  IO 

IOO 

•79 

+  .05 

25 

.90 

+  .16 

256 

-8.93 

+  0.19 

0.0361 

Mean  —  8.74 

+  1.03  —  0.99 

[z/z/]  —  0.2756 

\y  =  2.02 

Taking  the  observations  as  of  equal  precision,  we  find  the  arithmetic 
mean  to  be  —  8.74.  This  is  the  most  probable  value  of  the  correction. 

The  residuals  v  are  found  by  subtracting  each  observed  value  from  the 
most  probable  value  according  to  the  relation 

V-M-v 
They  are  written  in  two  columns  for  convenience  in  applying  the  check 


The  exact  mean  being  —  8.74/T>  the  quantity  24T  X  21  =  4  should  be,  as  it  is,  the 
numerical  difference  between  the  +  and  —  residuals.  Hence  we  may  con- 
sider the  mean  to  be  correctly  found. 


DIRECT    OBSERVATIONS.  73 

Precision  of  the  Arithmetic  Mean.  —  The  degree 
of  confidence  to  be  placed  in  the  most  probable  value  of  the 
unknown  is  shown  by  its  mean-  square  or  probable  error. 

43.   (a)  BesseTs  Formula. 

If  we  knew  the  true  value  T  of  the  unknown,  and  conse- 
quently the  true  errors  4»4  •  •  •  we  should  have,  as  in  Art. 
20,  for  the  m.  s.  e.  of  an  observation, 


n 

But   we    have    only  the    most   probable  value    V  and    the 
residual   errors  i\,  ?'„    .   .   .    vn  instead  of  the  true    values 

T,  4,  4  .  .  .  4,.    Now, 

y-v1=M1=  r-  4 

V-vt  =  M^=  7--  4  (i) 


By  addition,  remembering  that  [?']=:  o, 

nV=nT-[J]  (2) 

Substitute  for  Fin  equations  (i)  and 

nvl  =  (n—  i)4  —  4—  •  •  • 

WT'.,=:          -4  -)-(«_   1)4-    .    .    . 

Squaring, 

wX'=(«-i)'4'  +          4'  +  -  •  .-2(«-i)44-.  -  - 
»v,-  =  4'  +  («  -  0*4'  +  •  •  -  -  2(«  -  04  Ja  -  •  •  • 

By  addition,  assuming  that  the  double  products  destroy 
each  other,  positive  and  negative  errors  being  equally 
probable, 


=  («- 

=  :  _t^ 
w  — 

which  gives  the  m.  s.  e.  of  an  observation. 


and  //'  =  :  _^  (3) 

w  —  i 


74  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Now,  from  Art.  19, 


••  /'„  = 


which  gives  the  m.  s.  e.  of  the  arithmetic  mean  of  n  observa- 
tions of  equal  precision. 

The  result,  //  —  -^  -  ,  might  have  been  inferred  a  priori, 
n  —  i 

For  the  series  of  residuals  i\,v^  .  .  .  found  from  the  arith- 
metic mean  Fof  the  observed  values  approximates  closely 
to  the  true  series  of  errors  J  from  which  the  law  of  error 
was  derived.  Hence  we  conclude  that  the  formula 

//  =  M  (4) 


would  be  a  close  approximation  to  the  m.  s.  e.  of  an  observa- 
tion. It  is,  however,  not  satisfactory,  from  the  fact  that  it 
ought  to  become  indeterminate  when  n  =  i,  which  it  does 
not.  For  when  «=  i,  z>  —  o,  and  unless  the  denominator  of 
(4)  is  equal  to  o,  //  would  be  equal  to  o;  that  is,  the  first 
observation  would  give  the  true  value  of  the  unknown, 
which  is  absurd.  Hence  we  should  expect  the  formula  to 
be  of  the  form 

,,«_. 

1 


n-l 


which  becomes  of  the  indeterminate  form  -g-  when  n=  i. 
44.  As  in  Art.  23,  we  may  show  from  the  expansion  of 

—  —  //)   that  the  square  of  the  m.  s.  e.  ofu9  =  —      -   is 

n  —  i       '  /  n  —  i 

equal  to  //  4/^_       We  have,  therefore, 

r    n  —  i 


DIRECT    OBSERVATIONS.  75 

and  when  n  is  very  large 

/  [w]    /  I         \ 

'-V^'V  z:-fi&=$ 

that  is,  the  mean  uncertainty  of  /*  is 

\'2(n—  T) 

45.  From  the  constant  relation  existing  between  the 
m.  s.  e.  and  p.  e.  given  in  Art.  22  we  have  for  the  p.  e.  of 
an  observation  and  of  the  arithmetic  mean  of  n  observations 
respectively, 

r  —  p  4/2  n 


where  />  1/2  =  0.6745  nearly. 

46.  If  we  consider  the  m.  s.  e.  <j.0  of  the  arithmetic  mean 
of  ;/  observations  as  the  true  error  of  the  mean  V,  the  results 

fa—-£-  and  u*=-±      !-  may  be  derived  very  neatly  as  fol- 

\'n  n~l 

lows  : 

We  have 


By  addition,  taking  the  mean 


?6  THE  ADJUSTMENT   OF  OBSERVATIONS. 

Error  of  F=I  (4  +  4  +  •  •  •  +  4) 
and  ^='(4  +  4+...  +4)' 


But  from  Art.  20 

,_ 
7 


— 

Vn 
Again, 

A=T—M 


—  V+JL-.M 

'   Vn 


and 


Hence,  remembering  that  [v~\  =  o, 


47.  (b)  Peters  Formula. 

The  m.  s.  e.  and  p.  e.  of  a  series  of'  observed  values  may 
be  more  rapidly  computed  from  the  sum  of  the  errors  rather 
than  from  the  sum  of  their  squares  by  means  of  the  con- 
venient formula  first  given  by  Dr.  Peters.* 

From  the  equation 


*  Astronomische  Nachrichtin,  No.  1034. 


DIRECT   OBSERVATIONS.  77 

we  have  approximately,  without  regard  to  sign, 


Adding  and  dividing  by  ;/, 


But  from  Art.  22 


1.2533 
=     .        — =  \y    nearly. 

For  a  demonstration  of  this  formula  more  rigorous  in  form 
see  Astron.  Nachr.,  No.  2039. 

As  in  Art.  23,  it  follows  that  the  precision  of  this  formula 
is  expressed  by  the  complete  form 


1-2533 


the  last  term  giving  the  mean  uncertainty  of  /*. 

For  the  p.  e.  of  an  observation  and  of  the  arithmetic  mean 
of  «  observations  we  have  respectively 

0-8453 


- 

Vn(n  — 

0.8453 

r0  =  —  -  --  \y 


The  mode  of  deriving  Peters'  formula  given  in  the  pre- 
ceding is  approximate,  and  the  formula  itself  is  not  very 


/8  THE   ADJUSTMENT   OF   OBSERVATIONS. 

close  for  small  values  of  n.  Thus  when  n  =  2,  if  ^/denotes  the 
difference  of  the  observed  values,  then  by  Bessel's  formula 

d  d 

I*  =  —^,  and  by  Peters'  formula  it  =  1.25  —-=.,  which  is  one- 
V2  V2 

fourth  greater.  The  corresponding  probable  errors  are  in 
the  same  ratio. 

A  formula  for  the  probable  error  of  the  mean  which 
answers  better  than  Peters'  for  small  values  of  n  has  been 
derived  by  Fechner  (Poggendorff,  Annalen,  Jubelband,  1874) 
as  follows  : 

r  _         I-I9SS      ,   \v_ 

T°  ~  A/2  n  —  0.8548     n 

As  this  formula  is  troublesome  to  compute,  and  as  it  gives 
results  agreeing  closely  with  those  found  from  Peters' 
formula  when  n  is  a  moderately  large  number,  there  is  no 
advantage  to  be  derived  from  the  use  of  it  in  ordinary 
work. 

48.  Collecting  the  formulas  for  finding  the  p.  e.  of  a 
sinle  observation  and  of  the  arithmetic  mean  of  n  observa- 


tions, we  have 


r  =  0.6745  yl.  r  =  0.8453  -7= 

n  —  i  vn(n  —  i) 


—    /->      Q    A    f   1     


r0=  0.6745  y    !u_l}         r"=a8453,,v;/_  L 

To  save  labor  in  the  numerical  work  I  have  computed 
tables  containing  the  values  of  the  coefficients  of  V[^']  and 
[v  in  these  equations  for  values  of  n  from  2  to  100.  (See 
Appendix,  Tables  II.,  III.) 

If  Bessel's  formula  is  used  compute  first  [vv~],  then  V[^;] 
can  be  taken  from  a  table  of  squares  closely  enough.  This 
square-root  number  multiplied  by  the  number  in  Table  II.. 
corresponding  to  the  given  value  of  n  gives  the  p.  e.  sought. 
If  Peters'  formula  is  used  multiply  the  sum  of  the  residuals, 
without  regard  to  sign,  by  the  numbers  in  Table  III.  cor- 
responding to  the  argument  n. 


DIRECT   OBSERVATIONS.  79 

49.  Control  of  [?'?']•  —  A  control  is  afforded  by  the 
derivation  of  [i'?>\  from  the  observed  values  and  the  arith- 
metic mean  directly. 

We  have 


vn  =  V-MH 
Square  and  add, 

[w]  =  n  F2  -  2  F[J/  ]  +  [>/'] 

=  [j/']_[j/]r.  (i) 

since   nV=Jf 


The  values  of  J/2  may  be  found  from  a  table  of  squares 
or  from  Crelle's  tables,  or,  if  the  numbers  M  are  large,  an 
arithmometer,  or  machine  for  multiplying-  and  dividing,  may 
be  employed  with  advantage. 

The  computation  may  often  be  much  abbreviated  by  the 
artifice  of  Art.  41.  Substituting  the  values  of  J/p  M^  .  .  .  Mn 
from  that  article  in  (i),  we  find,  after  a  simple  reduction, 


=  [//]-[/>' 

50.  Approximate  Method  of  Finding  the  Pre- 
cision. —  A  connection  between  the  p.  e.  of  a  single  observa- 
tion and  the  greatest  error  committed  in  the  series  may  be 
established  approximately  by  the  aid  of  the  principle  proved 
in  Art.  30.  There  we  saw  that  in  a  large  series  the  actual 
errors  may  be  expected  to  range  between  zero  and  4  or 
5  times  the  p.  e.  of  an  observation.  If,  then,  we  find  from 
the  observations  a  p.  e.  of  an  amount,  say,  r,  we  may  assert 
that  the  greatest  actual  error  is  not  likely  to  be  more  than 
5r.  The  probability  of  its  being  as  large  as  this  is  only 
about  ToVo. 

The  same  principle  will  enable  us  to  estimate  roughly 
the  p.  e.  in  a  series  of  observations.  A  glance  at  the 


80  THE   ADJUSTMENT   OF   OBSERVATIONS. 

measured  results  will  show  the  largest  and  smallest,  and 
their  difference  may  be  taken  as  the  range  in  the  results, 
and  half  the  difference  as  the  maximum  error.  Hence, 
since  in  an  ordinary  series  of  from  25  to  100  observations 
the  maximum  error  may  be  expected  to  be  from  3  to  4  times 
the  p.  e.,  we  may  take  the  p.  e.  to  be  from  \  to  \  of  the  range 
of  the  errors  of  observation. 

This  result   may  be  confirmed   as  follows  :     Expanding 
the  exponential  function  <p(d)  in  a  series,  we  may  write 


—  P—  QJ?  -f  RJ  -  .  .  . 

where  P,  Q,  R,  .  .  .  are  constants. 

If  a  denotes  the  maximum  error,  then,  since  the  proba- 
bility of  the  occurrence  of  an  error  between  the  limits 
-\-a  and  —  a  is  certainty,  we  have,  taking  two  terms  of  the 
series, 


f 

j  -< 


Also,  since  a  is  the  maximum  error,  its  probability  is  zero. 


From  these  two  equations  P  and  Q  may  be  found  in  terms 
of  a. 

To  find  the  p.  e.  r  we  have  by  definition  (see  Art.  21). 


or  Pr  —  -  Qr>  —  - 

3  2 

Substituting  for  P  and  Q  their  values,  and  solving  for  r,  we 
find 

a  , 

r  — -  nearly 

3 

that  is,  the  p.  e.  is  approximately  \  of  the  maximum  error,  or 
£  of  the  range  of  the  errors  of  observation. 


DIRECT  OBSERVATIONS.  8 1 

A  closer  approximation  would  be  found  by  taking  three 
terms  of  the  series  for  ^(J).  We  should  then  find 

r  =  -  nearly. 

4 

See  note  by  Capt.  Basevi,  R.E.,  in  G.  T.  Survey  of  India, 
vol.  iv. ;  also  Helmert  in  Zeitschr.fiir  Verviess.,  vol.  vi. 

Ex,  We  shall  now  apply  the  preceding  formulas  to  the  example  in  Art. 
42  to  find  the  m.  s.  e.  and  p.  e.  of  the  arithmetic  mean  and  of  a  single 
observation. 

(i)   The  m.  s.  e.  and  p.  e,  of  the  arithmetic  mean. 

These  we  may  find  in  two  ways : 

(a)  From  the  sum  of  the  squares  of  the  residuals  (Art.  43) : 


ft0  =  |/_If^L 

'    n(n  —  i) 


21  X  2O 

=  0.026 
ra  =  0.6745  X  O.O26 

=  0.017 
or  from  Table  II.  at  once  : 

;-„  =0.525  X  0.033 

=  0.017 

(b)  From  the  sum  \v  of  the  residuals  (Art.  47): 

The  multiplier  in  Table  III.  corresponding  to  the  number  21  is  0.009. 
.  • .  ra  =  2.02  X  0.009 

=  o.oiS 

(2)   The  p.  e.  of  a  single  observation. 
From  Tables  II.  and  III.  directly: 

r  =  o. 525  X  0.151=  0.079 
r=2.O2    X  0.041  =0.082 

Check  («).  Let  the  residuals  be  arranged  in  order  of  magnitude.    They  are: 
0.23    0.22    0.19    0.16    0.16    o.n    o.n    o.io    o.io    o.io    0.09 
0.09    0.06    0.06    0.05    0.04    0.04    0.04    0.04    0.02    o.oi 

The  residual  0.09  occupies  the  middle  place,  and  is  therefore  the  p.  e.  r  of  a 
single  result  (Art.  21).     The  computation  above  gives  0.08. 
Check  (ft).  See  Art.  50. 

Range  =  0.22  +  0.23  =  0.45 

.-.r  =  °-i-5=o.oS. 
6 

The  values  found  by  the  different  methods  agree  reasonably  well. 


82 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


51.  The  Law  of  Error  Tested  by  Experience. — We 

shall  now  test  our  example  and  see  how  closely  it  conforms 
to  the  law  of  error,  and  hence  be  in  a  better  position  to 
judge  of  how  far  the  law  of  error  itself  is  applicable  in 
practice.  This  is  the  h.  posteriori  proof  intimated  in  Art.  11 
as  necessary  for  the  demonstration  of  the  law. 

(1)  The  number  of  -(-  residuals  is  12,  and  the  number  of 

-  residuals  is  9. 

(2)  The  sum  of  the  -f-  residuals  is  1.03,  and  the  sum  of  the 

-  residuals  is  0.99. 

(3)  The   sum  of  the  squares  of  the  -f-  residuals  is   1417, 
and  of  the  —  residuals  is  1339. 

(4)  The  p.  e.  of  a  single  observation  is  0.08.     To  find  the 
number  of  observations  we  should  expect   whose  residual 
errors  are  not  greater  than  o.  10,  we  enter  Table  I.  with  the 

argument  — ' — —  1.25  and  find  0.60.     This  multiplied  by  21 
0.08 

gives  13  as  the  number  of  errors  to  be  expected  not  greater 
than  o.  10.  By  actual  count  we  find  the  number  observed 
to  be  14. 

To  find  the  number  to  be  expected  between  o.  10  and  0.20 

2O 

we  enter  the  table  with  the  argument  --— —  2. 50  and  find  0.91. 

.08 

From  this  deduct  0.60  and  multiply  the  remainder  by  21. 
This  gives  6.  The  number  observed  is  5. 

The  number  to  be  expected  over  0.20  is,  by  theory,  2. 
The  number  observed  is  2. 

The  preceding  results  are  collected  in  the  following 
table  : 


Limits  of  Error. 

Number  of  Errors. 

Theory. 

Observation. 

s.                s. 

o.oo  to  o.  10 

13 

14 

0.10   tO   0.20 

6 

5 

over  o.  20 

2 

2 

DIRECT   OBSERVATIONS. 


Table  I.,  it  will  be  remembered,  is  founded  on  the  sup- 
position that  the  number  of  observations  in  a  given  set  is 
very  large.  In  our  example  the  number  is  only  21.  Per- 
fect accordance  between  the  number  of  errors  given  by 
theory  and  the  number  given  by  observation  is,  therefore, 
not  to  be  expected.  For  longer  examples  of  this  kind  see 
Chauvenet's  Least  Squares,  p.  489 ;  Airy's  Theory  of  Errors 
of  Observations,  third  ed.,  appendix. 

Comparisons  between  the  number  of  errors  within  given 
limits  that  actually  occur  in  a  series  of  observations  and  the 
number  to  be  expected  from  theory  in  the  same  series  show 
the  degree  of  confidence  we  may  place  in  the  law  of  error. 
It  is  the  final  criterion,  and  forms  the  second  part  of  the 
proof  of  the  law  as  stated  in  Art.  11. 

The  law  of  error  has  been  so  thoroughly  tested  in  this 
way,  so  far  as  the  sciences  of  observation  (for  which,  in- 
deed, it  was  framed)  are  concerned,  that  if  in  a  series  of 
observations  we  find  that  the  errors  do  not  conform  to  it 
we  may  suspect  the  presence  of  other  than  accidental 
sources  of  error.  For  example,  Bessel  *  found,  from  his  re- 
duction of  a  series  of  300  observations  made  by  Bradley  on 
the  declinations  of  a  Tauri,  etc.,  that  the  numbers  ot  errors 
that  actually  occurred  and  the  numbers  given  by  theory 
within  specified  limits  were  as  follows: 


Limits. 

Number  of  Errors. 

Experience. 

Theory. 

o".o  to  o" 

4 

66 

65 

o".4  to  o" 

S                             5S 

to 

O\8    [0    l" 

2 

55 

53 

I    .2    tO    l" 

6 

28 

41 

i".6  to  2" 

o                            27 

30 

2   .0   to    2 

4                             23 

21 

2".  4  to  2" 

S                            10 

13 

2".S  to  3" 

2                                     15 

S 

3".  2  to  3' 

6 

5 

3"-6  to  4" 

o                              4 

2 

over  4 

6 

- 

*  f'urti/antentii  As(r<>n<>>Nitr,  pp.  19,  20. 


84  THE  ADJUSTMENT  OF   OBSERVATIONS. 

The  agreement,  especially  for  the  larger  errors,  is  not 
very  close.  Prof.  Safford,  to  whom  I  referred  this  ex- 
ample, states  that  in  a  new  reduction  of  Bradley's  observa- 
tions made  by  Auwers  a  much  better  agreement  between 
experience  and  theory  is  found.  Three  sources  of  sys- 
tematic error  enter  into  the  observations  which  were  not 
taken  into  account  in  Bessel's  reduction  : 

(a)  Personal  equation.     Bradley  was  not  the  only  ob- 
server, Mason  and  Green  being  the  others. 

(b)  Local  deviation  of  the  plumb-line. 

(c)  The  assumption  that    the   collimation  of  Bradley's 
Quadrant  did  not  vary  irregularly. 

Auwers  has  in  a  great  degree  overcome  these  difficulties 
in  his  reduction.  He  finds,  for  example,  the  number  of 
errors  over  4"  to  be  2  instead  of  6,  thus  agreeing  with  the 
theoretical  number. 

For  another  illustration  see  Helmert's  discussion  of  the 
errors  in  Koppe's  triangulation  for  determining  the  axis  of 
the  St.  Gothard  Tunnel.  (ZeitscJir.  fur  Vermess,,  vol.  v. 
pp.  146  seq.) 

52.  As  the  Gaussian  law  of  error  is  found  to  apply 
reasonably  well  to  other  phenomena,  such,  for  example,  as 
statistical  questions,  guesses,  etc.,  it  has  been  often  rashly 
assumed  to  be  of  universal  application  ;  and  when  prediction 
and  experience  are  found  not  to  agree,  the  validity  of  the 
law  in  any  case  has  been  as  rashly  impugned. 

In  the  fundamental  investigation  in  Art.  15  the  hypoth- 
eses there  made  are  satisfied  by  other  functions  of  the 
measured  values  besides  the  arithmetic  mean.  Thus,  tor 
example,  taking  the  geometric  mean  g,  the  resulting  law  of 
error  would  be  of  the  form 


where  c  and  h  are  constants. 

This  form  appears  to  apply  to  many  statistical  questions 
better  than   the    Gaussian  law.     That   it   does  so  is,  how 

*  Gallon,  Proc.  Roy,  Sac.  Land.,  1879,  pp.  365  seq. 


DIRECT   OBSERVATIONS.  85 

ever,  no  argument  against  the  Gaussian  law  in  its  own 
territory. 

A  curious  illustration  of  the  preceding  remarks,  in  the 
misapplication  of  the  Gaussian  law  of  error  to  a  case 
where  it  would  appear  at  first  sight  that  it  ought  to  apply 
directly,  is  to  be  found  in  the  essay  on  "  Target-Shoot- 
ing "  in  Sir  J.  F.  \V.  Herschel's  Familiar  Lectures  on  Scientific 
Subjects. 

53.  Caution  as  to  the  Application  of  the  Tests 
of  Precision. — In  the  preceding  article  we  have  given 
several  cautions  with  regard  to  the  strict  application  of  the 
law  of  error  in  practice.  We  shall  now  perform  a  similar 
service  for  the  tests  of  precision — the  m.  s.  e.  and  the  p.  e. 

The  m.s.  e.  and  p.  e.  of  series  of  observations  have  been 
defined  as  measures  of  their  relative  accuracy.  With  ideal 
series — that  is,  series  which  do  not  contain  systematic 
errors,  and  in  which  the  accidental  errors  are  continuous  in 
magnitude  between  the  extreme  limits  of  error — this  is  true  ; 
and  in  actual  series,  in  good  work,  it  is  on  the  whole  true. 
But  in  any  actual  series  selected  at  random  we  must  apply 
these  tests  with  caution.  It  is  a  common  mistake  to  over- 
look  the  distinction  between  observations  which  conform  to 
the  law  of  error  and  those  which  only  apparently  do  so,  and 
hence  to  condemn  the  m.  s.  e.  and  p.  e.  as  not  only  worth- 
less but  misleading. 

For  example,  in  levelling,  if  the  same  line  is  run  over  in 
duplicate  in  the  same  direction,  a  good  agreement  may  be 
expected  at  the  several  bench-marks  where  comparisons 
are  made.  The  m.  s.  e.  of  observation  will  consequently  be 
small.  If  the  line  is  levelled  in  opposite  directions  experi- 
ence shows  that  the  agreement  would  not  be  so  good.  The 
m.  s.  e.  would  be  larger  than  before.  We  might,  therefore, 
hastily  conclude  that  the  first  work  would  give  the  better 
result.  But  when  we  reflect  that  the  main  differences  arise 
from  such  causes  as  the  refraction  of  light,  the  personal  bias 
of  the  observer,  etc.,  which  causes  are  less  likely  to  be 
mutually  destructive  and  more  likely  to  be  cumulative  if 


86  THE   ADJUSTMENT   OF   OBSERVATIONS. 

the  lines  are  run  in  the  same  direction,  it  is  to  be  expected 
that  the  final  result  obtained  from  measurements  in  opposite 
directions  will  be  nearer  the  truth.  The  conclusion  arrived 
at  by  trusting  to  the  m.  s.  e.  alone  would  be  illusory. 

Again,  in  measuring  a  horizontal  angle,  if  the  same  part 
of  the  limb  of  the  instrument  is  used  in  making  the  readings 
the  results  may  be  very  accordant  and  the  m.  s.  e.  conse- 
quently small,  but  the  angle  itself  may  not  be  anywhere 
near  the  truth.  This  would  be  shown,  for  example,  by 
the  large  discrepancies  in  the  sums  of  angles  of  triangles 
measured  in  this  way  from  the  theoretical  sums.  For  a 
long  time  this  contradiction  was  a  source  of  much  per- 
plexity, and  many  good  instruments  were  most  unjustly 
condemned.*  At  last  the  discovery  was  made  that  it  was 
mainly  owing  to  periodic  errors  of  graduation  of  the  limb 
which,  when  corrected  for,  made  the  remaining  errors  fairly 
subject  to  the  law  of  error.  This  most  important  discovery 
may  be  said  to  have  revolutionized  the  art  of  measuring 
horizontal  angles. 

The  difficulty  may  be  explained  in  this  way.  In  the 
derivation  of  the  m.  s.  e.  from  a  series  of  n  observed  quanti- 
ties Mlf  M^  .  .  .  we  had  the  observation  equations 


\vv\ 
Also  //  = 


n  —  i 

Now,  if  we  suppose  each  of  the  observed  quantities  to  be 
changed  by  the  same  amount  c,  which  may  be  of  the  nature 
of  a  constant  error  or  correction,  so  that  they  become 
M^-{-ct  M^-\-c,  .  .  .  the  most  probable  value,  instead  of 
being  V,  will  be  'V-\-c.  Also  since 


=  V-M 
the  residual  errors  will  be  the  same  as  before. 

*  See,  for  example,  G.  T.  Survey  of  India^  vol.  ii.  pp.  51,  96. 


DIRECT   OBSERVATIONS.  87 

Hence  if  is  unchanged,  and  we  see,  therefore,  that  the 
m.  s.  e.  makes  no  allowance  for  constant  errors  or  correc- 
tions to  the  observed  quantity.  These  are  supposed  to  be 
eliminated  or  corrected  for  before  the  most  probable  value 
and  its  precision  are  sought. 

Another  common  misapprehension  is  the  following: 
From  Arts.  19  or  46  the  relation  between  the  m.  s.  e.  ol  a 
single  observation  n  and  the  m.  s.  e.  of  the  mean  of  n  obser- 
vations 0  is 


This  formula  shows  that  by  repeating  the  measurement  a 
sufficient  number  of  times  we  can  make  the  m.  s.  e.  of  the 
final  result  as  small  as  we  please.  Nothing  would,  there- 
fore, seem  to  be  in  the  way  of  our  getting  an  exact  result, 
and  that  we  could  do  as  good  work  with  a  rude  or  imper- 
fect instrument  as  with  a  good  one  by  sufficiently  increasing 
the  number  of  observations. 

Experience,  however,  shows  that  in  a  long  series  of 
measurements  we  are  never  certain  that  our  result  is  nearer 
the  truth  than  the  smallest  quantity  the  instrument  will 
measure.  If  an  instrument  measures  seconds  we  cannot  be 
sure  that  by  repeating  the  observations  we  can  get  the 
nearest  hundredth  or  tenth.  In  a  word,  we  cannot  measure 
what  we  cannot  see. 

Take  an  example:  With  the  meridian  circle  Prof.  Rogers 
found  the  p.  e.  of  a  single  complete  observation  in  declina- 
tion to  be  ±  o".36,  and  the  p.  e.  of  a  single  complete  observa- 
tion in  right  ascension  for  an  equatorial  star  to  be  +  os.O26. 
He  says:  "  If,  therefore,  the  p.  e.  can  be  taken  as  a  measure 
of  the  accuracy  of  the  observations,  there  ought  to  be  no 
difficulty  in  obtaining  from  a  moderate  number  of  observa- 
tions the  right  ascension  within  os.O2  and  the  declination 
within  o".2.  Yet  it  is  doubtful,  after  continuous  observations 
in  all  parts  of  the  world  for  more  than  a  century,  if  there  is 
a  single  star  in  the  heavens  whose  absolute  co-ordinates  are 


88  THE  ADJUSTMENT   OF   OBSERVATIONS. 

known  within  these  limits."  The  explanation  is,  as  inti- 
mated, that  constant  errors  are  not  eliminated  by  increasing 
the  number  of  observations.  Accidental  errors  are  elimi- 
nated by  so  doing  ;  and  if  a  number  of  observations  ex- 
pressed by  an  infinity  of  a  sufficiently  high  order  could  be 
taken,  so  that  the  constant  errors  entering  in  the  different 
series  could  be  classed  as  accidental,  these  errors  would 
mutually  balance  in  the  reduction  and  we  should  arrive  at 
the  true  result. 

Closely  allied  to  the  preceding  is  the  common  idea  that 
if  we  have  a  poor  set  of  observations  good  results  can  be 
derived  from  them  by  adjusting  them  according  to  the 
method  of  least  squares,  or  that  if  work  has  been  coarsely 
done  such  an  adjustment  will  bring  out  results  of  a  higher 
grade.  A  seeming  accuracy  is  obtained  in  this  way,  but  it 
is  a  very  misleading  one.  The  method  of  least  squares  is 
no  philosopher's  stone:  it  has  no  power  to  evolve  reliable 
results  from  inferior  work. 

A  third  source  of  uncertainty  from  the  same  cause  may 
be  mentioned.  It  may  happen  that  the  value  obtained  of 
the  p.  e.  is  numerically  greater  than  that  of  the  observed 
quantity  itself.  It  is  then  a  question  whether  in  subsequent 
investigations  we  should  use  the  value  of  the  observed 
quantity  as  found  or  neglect  it.  This  depends  on  circum- 
stances. It  is  ever  a  principle  in  least  squares  to  make  use 
of  all  the  knowledge  on  hand  of  the  point  at  issue.  If  we 
have  strong  a  priori  reasons  for  expecting  the  value  zero  it 
would  be  better  to  take  this  value.  Thus  if  we  ran  a  line 
of  levels  between  two  points  on  the  surface  of  a  lake  we 
should  expect  the  difference  of  height  to  be  zero.  If  the 
p.  e.  of  the  result  found  were  greater  than  the  result  itself 
it  would  be  allowable  in  this  case  to  reject  the  determina- 
tion. On  the  other  hand,  when  we  have  no  a  priori  know- 
ledge, as  in  determinations  of  stellar  parallax,  for  example, 
if  the  p.  e.  of  the  value  found  were  in  excess  of  the  value 
itself,  as  is  sometimes  the  case,f  we  could  do  nothing  but 

*  Proc.  Amcr.  Acad,  Sci.,  1878,  p.  174.      tSee,  for  example,  Newcomb,  Astronomy,  app.  vii. 


DIRECT   OBSERVATIONS.  89 

take  the  value  resulting  from  the  observations,  unless,  in- 
deed, it  came  out  with  a  negative  sign,  and  then  its  unre- 
liable character  would  be  evident. 

54.  Constant  Error. — The  remarks  on  constant  error 
in  the  preceding  articles  lead  us  to  notice  an  example  or 
two  of  the  detection  and  treatment  of  this  great  bugbear  of 
observation. 

We  suspect  the  presence  of  constant  errors  in  a  series  of 
observations  from  the  large  range  in  the  results — a  range 
greater  than  would  naturally  be  expected  after  all  known 
corrections  have  been  applied.  Great  caution  is  necessary 
in  dealing  with  such  cases,  and  one  should  be  in  no  hurry  to 
jump  at  conclusions. 

Sometimes  the  sources  of  error  are  detected  without 
much  trouble.  Thus  in  measuring  an  angle  with  a  theod- 
olite, if  the  instrument  is  placed  on  a  stone  pillar  firmly 
embedded  in  the  ground,  the  range  in  results,  if  targets  are 
the  signals  pointed  at,  would  not  usually  be  over  10"  in 
primary  work  ;  and  on  reading  to  a  number  of  signals  in 
order  round  the  horizon  the  final  reading  on  closing  the 
horizon  would  be  nearly  the  same  as  the  initial  reading  on 
the  same  signal.  If  next  the  instrument  were  placed  on  a 
wooden  post  or  tripod,  and  readings  made  to  signals  in 
order  round  the  horizon  in  the  same  way  as  before,  the 
final  reading  might  differ  from  the  initial  by  a  large  amount. 
The  observations  might  also  show  that  the  longer  the  time 
taken  in  going  around  the  greater  the  resulting  discrepancy. 
The  natural  interence  would  be  that  in  some  way  the 
wooden  post  had  to  do  with  the  discrepancy  in  the  results. 
In  an  actual  case*  of  this  kind  examination  showed  the 
change  to  be  most  uniform  on  a  clav  when  the  sun  shone 

t> 

brightlv.      Measurements  were   then   made   at   night,  using 

*  At  U.  S.  Lake  Survey  station  Emle1,  Lake  Superior,  many  observations  were  taken  during 
both  day  and  night  in  July,  1871,  to  determine  the  rate  of  twi-t  of  centre-post  on  which  the  theod- 
olite used  in  measuring  angles  was  placed.  The  conclusion  arrived  at  was  that  "  during  a  day  of 
uniform  sunshine  and  clear  atmosphere  this  twist  seemed  to  be  quite  regular,  and  at  the  rate  of 
about  ant  second  of  arc  per  minute  of  ti»te,  reaching  a  maximum  about  7  P.M.  and  a  minimum 
about  7  A.M.,  during  the  month  of  July.  On  partially  cloudy  days  there  was  no  regularity  in  the 
twist,  being  sometimes  in  one  direction  and  again  in  the  opposite.'' 


9O  THE   ADJUSTMENT   OF   OBSERVATIONS. 

lamps   as   signals   on    the    distant    stations,    and    the    same 
change  was  observed,  only  it  was  in  the  opposite  direction. 

The  effect  on  the  value  of  an  'angle  of  this  twist  of  sta- 
tion, assuming  it  to  act  uniformly  in  the  same  direction 
during  the  time  of  observation,  can  be  eliminated  by  the 
method  of  observation  :  first  reading  to  the  signals  in  one 
direction  and  then  immediately  in  the  opposite  direction, 
and  calling  the  mean  of  the  difference  of  the  two  sets  of 
readings  a  single  value  of  the  angle.  'So  also  in  azimuth 
work  the  mean  of  the  difference  of  the  readings,  star  to 
mark,  and  mark  to  star,  gives  a  single  value  free  from 
station-twist. 

This  mode  of  procedure  is  in  accordance  with  the  general 
principle  to  eliminate  a  correction,  when  possible,  by  the 
method  of  observation,  rather  than  to  compute  and  apply  it. 
See  Art.  2. 

The  effort  to  avoid  systematic  error  causes  in  general  a 
considerable  increase  of  labor,  and  sometimes  this  is  very 
marked.  For  example,  in  the  micrometric  comparison  ol 
two  line  measures  belonging  to  the  U.  S.  Engineers  the 
results  found  by  different  observers  showed  large  dis- 
crepancies. The  micrometer  microscopes  used  were  of  low 
power,  with  a  range  of  about  one  mm.  between  the  upper 
and  lower  limits  of  distinct  vision.  Examination  showed 
that  the  discrepancies  arose  mainly  from  focusing,  each  ob- 
server's results  being  tolerably  constant  for  his  own  focus. 
As  the  value  of  a  revolution  of  the  micrometer  screw 
entered  into  the  reduction  of  the  comparison  work,  and  as 
this  value  was  obtained  from  readings  on  a  space  of  known 
value,  error  of  focusing  entered  from  this  source.  Hence  a 
value  of  the  screw  had  to  be  determined  from  a  special  set 
of  readings  taken  at  each  adjustment,  and  this  value  used 
in  reducing  the  regular  observations  made  with  the  same 
focus.  Had  the  microscopes  been  of  high  power  it  would 
have  been  sufficient  to  have  determined  the  value  of  the 
screw  once  for  all,  since  the  error  arising  from  change  of 
focus  could  have  been  classed  as  accidental. 


DIRECT   OBSERVATIONS.  9  1 

In  trying  to  avoid  systematic  error  the  observer  will,  as 
he  gains  in  experience,  take  precautions  which  would  at 
first  seem  to  be  almost  childish.  Good  work  can  only  be 
had  at  the  cost  of  eternal  vigilance. 

55.  Necessary  Closeness  of  Computation.  —  The 
number  of  observations  necessary  to  the  proper  determina- 
tion of  a  quantity  may  be  approximated  to  by  referring  to 
the  m.  s.  e.  of  these  observations.  If,  after  planning  the 
method  of  observing  with  the  view  of  eliminating  "con- 
stant error,"  we  find  that  increasing  the  number  of  obser- 
vations beyond  a  certain  limit  does  not  sensibly  affect  the 
m.s.  e.  ,  we  ma}'  conclude  that  we  have  a  sufficient  number. 
It  is  evident  that  increasing  or  decreasing  this  number  will 
affect  the  result  more  or  less.  But  we  cannot  say  that  we 
are  nearer  the  truth  in  either  case. 

Hence  the  folly  of  a  too  rigorous  computation.  An 
approximate  value  being  the  best  that  we  can  have  at  any 
rate,  no  weight  is  added  to  it  by  carrying  out  that  value  to 
a  great  many  decimal  places.  Thus  if  the  most  probable 
value  of  a  quantity  computed  in  an  approximate  way  is  V, 
whereas  the  value  found  from  the  same  observations  by  a 
rigorous  computation  is  V-\-c,  we  may  estimate  the  allow- 
able value  of  c  as  follows.  Let  Jp  J3,  .  .  .  Jn  be  the  errors 
of  F,  then  Jl-\-  c,  J^-\-c,  .  .  .  Jn-\-c  are  the  errors  of  V-\-c. 
Hence 


and 

/V+c=#p|i-|  —  --|  approxi  match'. 

V      2  /v/ 

Now,  we  may  safely  allow  the  difference  between  //r  .  ,  and 
13 


92  THE   ADJUSTMENT   OF   OBSERVATIONS. 


Hv  to  be  -£-^.     Hence  there  will  be  no  appreciable  error  in- 
troduced by  computing  in  such  a  way  that 

--  <  — 

2  fiy      "  100 

that  is,  when  the  error  c  committed  is,  roughly,  ^  of  the 
m.  s.  e. 


B.  Observed  Values  of  Different  Quality. 

56.  The  Most  Probable  Value:  the  Weighted 
Mean.  —  It  has  been  shown  in  Art.  17  that  if  the  directly 
observed  values  M^  Mv  .  .  .  Mn  of  a  quantity  are  of  dif- 
ferent quality,  the  most  probable  value  is  found  by  multiply- 
ing each  residual  error  of  observation  by  the  reciprocal  of 
its  m.  s.  e.,  and  making  the  sum  of  the  squares  of  the  pro- 
ducts a  minimum  ;  that  is,  with  the  usual  notation, 

—*  +  —..  !+•   -   •  +^i  =  amin.  (i) 

/V       /V  !*n 

or 


=a  min- 


By  differentiation  and  reduction, 

(3) 


We  have,  therefore,  the  equivalent  rule: 

If  the  observed  values  of  a  quantity  are  of  different  quality, 
the  mast  probable  value  is  found  by  multiplying  each  observed 
value  by  the  reciprocal  of  the  square  of  its  m.  s.  e. ,  and  dividing 
the  sum  of  the  products  by  the  sum  of  the  reciprocals. 

The  form  of  the  expression  for  V  suggests  another 
stand-point  from  which  to  consider  it.  Let  p^p»  .  .  .  /„ 


DIRECT   OBSERVATIONS.  93 

be  the  numerical  parts  of-  ,  such  that  each    is 

/v  /v      /v 

of  the  type 

(unit  of  measure)2 

P  — 

I? 

then  equation  (3)  may  be  written 


Also,  since  —  ,  —  ,  .  .  .  -  -  are  similarly  involved  in  the 

f*i    t**  P* 

numerator  and  denominator  of  the  value  of  V,  this  value 
will  remain  the  same  if/,,/.,,  .  .  .  /„  are  taken  any  num- 

bers whatever  in  the  same  proportion  to  —  ,  —  ,  .   .  . 

/V   /V  f'-n 

that  is,  if/>,'A»  •  •  •  Pn  satisfy  the  relations 


(5) 


where  //  is  an  arbitrary  constant. 

57.  Let  us  look  at  this  question  from  another  point  of 
view.  It  is  in  accordance  with  our  fundamental  assump- 
tions that  observations  of  a  quantity  made  under  the  same 
conditions,  so  that  there  is  no  a  priori  reason  for  choosing 
one  before  another,  are  of  the  same  quality.  They  require 
the  same  expenditure  of  time,  labor,  money,  etc.  If,  there- 
fore, we  represent  the  quality  of  a  single  observation  of  a 
certain  series  by  unity,  the  quality  of  the  arithmetic  mean 
of/  such  observations,  as  it  would  require  /  times  the  ex- 
penditure to  attain  it,  would  be  represented  by  />.  Let.  us 
suppose,  then,  that  the  arithmetic  mean  of  /,  observed 
values  of  a  certain  quality  is  J/,  ;  of  /.,  other  observed 
values  of  the  same  quality  it  is  J/,,  and  so  on.  The  total 
number  of  observed  values  is  [/»].  All  of  the  observed 
values  being  of  the  same  quality,  the  most  probable  value 


94  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Fof  the  unknown  is  given  by  the  arithmetic  mean;  that 
is,  by 

y  _  sum  of  the  values  of  all  the  sets 
~sum  of  number  of  obs.  in  each  set 


A+A+  •  •  •  +A 


which  is  of  the  same  form  as  Eq.  4,  above. 

The  numbers  A»  A  .....  pn  are  called  the  weights,  or, 
better,  the  combining  weights,  of  the  observed  values,  and 
the  mean  value  Fis  called  the  weighted  mean. 

In  view  of  this  definition,  the  general  principle  stated  in 
Art.  56  may  be  replaced  by  the  following  necessarily  equiva- 
lent one  : 

If  the  'observed  values  of  a  quantity  are  of  different  weights, 
the  most  probable  value  is  found  by  multiplying  the  square  of 
each  residual  error  of  observation  by  its  weight,  and  making  the 
sum  of  the  products  a  minimum. 

Thus  the  most  probable  value  V  is  found  from 

\_pvv\  =  a  min. 
that  is,  from 


By  differentiation, 

pl(V 
whence 


This  form  of  the  value  of  V  leads  to  the  rule  : 

If  the  observed  values  of  a  quantity  are  of  different  weights, 
the  most  probable  value  is  found  by  multiplying  each  observed 
value  by  its  weight,  and  dividing  the  sum  of  the  products  by 
the  sum  of  the  weights. 


DIRECT   OBSERVATIONS.  95 

As  in  the  case  of  the  arithmetic  mean  (Art.  40),  it  is  evi- 
dently simpler  in  practice  to  find  the  weighted  mean  directly 
by  this  rule  rather  than  from  the  minimum  equation. 

If  the  observed  values  J/are  numerically  large  we  may 
lighten  the  arithmetical  work  by  finding  V  by  the  method 
of  Art.  41.  Proceeding  as  there  indicated,  we  have 


=  X'  -\-  x"  suppose 

58.  Reduction  of  Observed  Values  to  a  Common 
Standard.  —  The  principle  of  the  weighted  mean  is  evi- 
dently an  extension  of  that  of  the  arithmetic  mean,  as  was 
pointed  out  long  ago  by  Cotes,  Simpson,  and  others.  It 
merely  amounts  to  finding  a  mean  of  several  series  of 
means,  the  unit  of  measure  being  the  same  in  each.  As 
soon,  therefore,  as  results  of  different  weights  are  changed 
into  others  having  a  common  standard  of  weight,  the  rules 
for  combining  and  finding  the  precision  of  observed  quantities 
of  the  same  weight  can  be  applied  to  weighted  quantities. 

This  change  we  are  enabled  to  make  by  means  of  the 
relation  (5),  Art.  56,  which  may  be  written 


— 

VA        A/A  VA 

Now,  since  /Jtt,  /*„  .  .  .  ttn  are  the  m.  s.  e.  of  Mlt  Mv  .  .  .  Mn, 
the  m.  s.  e.  of  M^p^M^/p^  .  .  .  J/wV/«  would  each  be  the 
same  quantity  n. 

Hence  if  a  serits  of  observed  values  Mt,  M^  .  .  .  Mn  have  tJie 
weights  A»  A>  •  •  •  A>  they  are  reduced  to  the  same  standard  by 
multiplying  by  A/A,  A/A>  •  •  •  VA  respectively. 

For  example,  given  the  observation  equations, 

V—  J/,  =vt  weight  A 
V-M,  =  i>,       "        A 

V-Mn=vn      "       A 
to  find  the  most  probable  value  of  V. 


g6  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Reducing  to  the  same  standard  of  weight,  we  have  the 
equations 

A/A  v-  VAM,  =  A/A^ 


Vpnv—  A/A,  ^4= 

and  the  most  probable  value  of  Fis  found  by  making 

(A/A  <0*  +  (A/A  fO*  +  -  •   •  +  (  VA  *„)'  =  a  rain. 
that  is,  by  making 

V-  A/A  ^) 


Reducing  this  equation,  \ve  find,  as  before, 


59.  Computation  of  the  Weights.  —  If  A>  A»  •  •  •  A 

represent  a  series  of  weights  corresponding  to  the  m.  s.  e. 
/^j,  //2,  .   .  .  /in,  then  we  have  the  relations 


where  the  value  of  //  is  entirely  arbitrary.  The  combining 
weights  are,  therefore,  known  when  the  m.s.e.  f^,/^,/^,  .../*„ 
are  known. 

These  relations  suggest  that  it  would  be  convenient  to 
define  //  as  the  m.  s.  e.  of  a  single  observation  assumed  to  be 
of  the  weight  unity. 

We  shall  define  11  in  this  way,  so  that  in  future  it  is 
understood  that  the  standard  to  which  observations  of  dif- 
ferent weights  are  reduced  for  comparison  and  combination 
is  the  fictitious  observation  whose  weight  is  unity  and 
whose  m.  s.  e.  is  /x 

60.  Control  of  the  Weighted  Mean.  —  Eq.  2,  Art.  57, 
may  be  written 

[pv]  =  o 


DIRECT   OBSERVATIONS.  97 

Hence  if  the  weighted  mean  V  has  been  computed  correctly, 
the  sum  of  the  products  of  each  residual  error  by  its 
weight  is  equal  to  zero. 

Usually,  however,  Fis  not  an  exact  quotient — that  is, 
[/J/]  is  not  exactly  divisible  by  [/>] — and  then  the  dis- 
crepancy of  [/7'J  from  zero  is  evidently  equal  to  the  prod- 
uct of  the  sum  of  the  weights  and  the  difference  between 
the  exact  value  of  Fand  the  approximate  value  used.  See 
Art.  42. 

Ex.  i. — Deduce  the  relation  [/^]  —  o  from  the  observation  equations 
directly. 

[Multiply  each  observation  equation  by  its  weight,  and  add  the  products.] 

Ex.  2. — Find  the  most  probable  value  of  the  velocity  of  light  from  the 
following  determinations  by  Fizeau  and  others: 

298000  kil.  ±  1000  kil. 
298500   "     ±  looo    " 
299990    "     ±    200   " 
300100   "     ±  1000    " 
299930    "     ±     100   " 

{Amer.  Jour.  Sci.,  vol.  xix.) 

The  weights,  being  inversely  as  the  squares  of  the  probable  errors,  are  as 
the  numbers  i,  i,  25,  i,  100.  (Art.  59.) 

(a)  Direct  solution. 


,  M             p 

pM 

298000 

i 

298000 

298500 

i 

298500 

299990 

25 

7499750 

300100 

i 

300100 

299930 

IOO 

29993000 

128 

38389350 

V •=  -j—. :-  =  299917  kil.  approx.. 


the  exact  value  being  2999 16||  kil. 


THE  ADJUSTMENT  OF  OBSERVATIONS, 
(b)  Solution  according  to  Art.  57. 

Assume  X'  =  298000 


/ 

/ 

pi 

o 

I 

o 

500 

I 

500 

1990' 

25 

49750 

2IOO 

I 

2100 

1930 

IOO 

I93OOO 

128 

245350 

and  V  =  298000  +  1916^  =  299916!^,  as  before. 

Control.  Take  V=  299917,  and  proceed  to  find  [/£'].     (See  Art.  60.) 


V 

P 

pv 

1917 

i 

1917 

1417 

i 

1417 

-  73 

25 

—  1825 

-183 

i 

-  183 

-  13 

IOO 

—  1300 

26 

The  discrepancy  should  be 

128  (299917  —  299916^)  =  26 
which  it  is. 


DIRECT   OBSERVATIONS.  99 

61.  The  Precision  of  the  Weighted  Mean.  —  Since 
the  weighted  mean  Fis  the  arithmetic  mean  of  [/>]  observa- 
tions of  the  unit  of  weight,  its  weight  is  [/].  Hence  the 
m.  s.  e.  Y  of  Fis  found  from 


where  n  is  the  m.  s.  e.  of  an  observation  of  the  unit  of  weight 
(standard  observation). 

According  to  Art.  58,  the  value  of  }t  may  be  found  by 
writing  \X/>,  7-,,  VA  ?'2>  .  .  .  for  vlt  i\,  ...  in  the  formulas 
derived  for  observations  of  the  same  weight.  Hence,  sub- 
stituting in  Bessel's  and  in  Peters'  formulas  Arts.  43,  47,  we 
have 


:=  1.2533 


n  —  i  \/n(n —  i) 

and  therefore 


..3533- 


These  expressions  reduce  to  those  for  the  arithmetic  mean 
where  the  observed  values  are  of  the  same  weight  by 
putting  [/]  =  up. 

62.  Control  of  \pi'v\.  —  A  control  of  the  accuracy  of  [/>?'?'] 
is  afforded  by  the  derivation  of  this  quantity  from  the 
observed  values  directly. 

The  observation  equations  are 

Vl—V—M,     weight/, 
vt=V-Mt         "      A 


Hence 


100  THE   ADJUSTMENT   OF   OBSERVATIONS. 

By  addition, 

\_pirj]  =  [>]  V*  - 


since 


In  using  this  formula  the  troublesome  term  is  \_pMM\ 
With  large  values  of  M  it  is  better  to  deduce  it  from  the 
column  pM)  already  computed  in  finding  the  weighted  mean. 
This  is  specially  advisable  if  one  has  a  machine  for  perform- 
ing multiplications.  With  small  values  of  M  a  table  of 
squares  is  best. 

With  large  values  of  M  we  may  perhaps  proceed  still 
more  conveniently  by  the  method  explained  in  Arts.  41,  57. 
With  the  same  notation  and  reasoning  as  there  employed  it 
is  found  that 


where  the  quantities  /are  numerically  small. 

Ex.  The  linear  values  found  for  the  space  o"«.oo  to  O»*.O5  of  inch  \ab\ 
on  the  standard  steel  foot  I  F.  of  the  G.  T.  Survey  ol  India  were  as  follows: 
o»".  050027,  o»«.O4997i,  o'».  050019,  o'«.  050079,  o1'".  050021,  o»».  050011.  The  num- 
bers of  measures  in  these  determinations  were  6,  6,  15,  15,  8,  8  respec- 
tively. 

Taking  the  numbers  of  measures  as  the  weights  of  the  respective  determi- 
nation-, required  the  most  probable  value  of  the  space  and  its  p.  e. 

The  direct  solution  presents  no  difficulty.  The  value  of  F  may  be  found 
as  in  Ex.  Art.  60,  and  thence  the  residuals  v.  The  m.  s.  e.  or  p.  e.  follows 
from  the  formulas  of  Art.  61. 

We  shall  give  the  solution  according  to  the  methods  of  Arts.  57  and  62, 
which  are  advantageous  in  this  case  on  account  of  the  large  numbers  that 
enter. 


DIRECT  OBSERVATIONS. 

Assume  X'  =  0.049971 


101 


/ 

P 

// 

/// 

56 

6 

336 

18816 

o 

6 

o 

o 

48 

15 

720 

34560 

108 

15 

1620 

1  74960 

50 

8 

400 

2OOOO 

40 

8 

320 

I2SOO 

58 

3396 

26II36  =  [///] 
198842  =  [//].r" 

62294  =  [pvv] 

=  0.6745  A/    62294 

r  58(6-1) 


V  —  0.049971  +  0.000059 
•=  0.050030 

V  =  O"*. 


=  o.ooooio 
o"«.c>oooio 


By  choosing  the  approximate  value  X'  equal  to  the  smallest  of  the  measures 
or  equal  to  the  greatest  of  them,  all  of  the  remainders  /  have  the  same  sign, 
which  is  a  great  convenience  in  computation. 

In  this  example  an  important  practical  point  occurs, 
and  one  often  overlooked.  The  p.  e.  is  not  computed  from 
the  original  observations,  but  from  these  observations 
grouped  in  six  sets  of  means.  These  means  we  have  treated 
as  if  original  observations  of  certain  weights.  Had  the 
original  observations  been  accessible  we  should  have  used 
them,  and  would  most  probably  have  found  a  different 
value  of  the  p.  e.  from  that  which  we  have  obtained.  This 
arises  from  the  small  number  of  observations  in  the  several 
sets.  In  good  \vork  the  difference  to  be  expected  between 
the  value  of  the  p.  e.  found  from  the  means  and  that  found 
from  the  original  observations  would  be  small.  Still,  when- 
ever there  is  a  choice,  the  p.  e.  should  always  be  deduced 


102  THE   ADJUSTMENT   OF   OBSERVATIONS. 

from  the  original  observations  rather  than  from  any  com- 
binations of  them. 

The  weighted  mean  value  V  would  evidently  be  the 
same  whether  computed  from  the  partial  means  or  from  the 
original  observations. 

Observed  Values  Multiples  of  the  Unknown. 

63.  Let  the  observed  values  Mlt  M^  .  .  .  MK  be  multi- 
ples of  the  same  unknown  X  '  ;  that  is,  be  of  the  form 
atX,  a^X,  .  .  .  aHX,  where  a^  av  .  .  .  an  are  constants  given 

by  theory  for  each  observation.     The  values  —  \  —  ?,  .  .  .  —  ? 

«,     #*  an 

of  X  may  be  regarded  as  directly  observed  values  of  unequal 
weight.      If  fi.  is   the  m.  s.  e.  of  an  observation,  that  is, 

of   M»MV  .  .  .,   then,    since    the    m.  s.  e.  of  _'  is  ^,of  —  ? 

al        a1         a^ 

is^,  .  .  .  the   weights  of  these   assumed  observations  are 
a, 

proportional   to  a*,  a*,  .  .  .      Hence   taking    the  weighted 
mean 


Also,  since  [V]  is  the  weight  of  X, 

*_    P* 
=  [«'J 

Cotes,*  in  solving  this  problem,  reasons  that,  since  for 
the  same  error  of  M  the  greater  a  is,  the  less  is  the  error  of 
X,  we  may  take  the  coefficients  a  to  express  the  relative 
weights  of  the  values  of  X.  Now,  placing  the  coefficients 
at,  a^  .  .  .  as  weights  along  a  straight  line  at  distances 

_',  ^?,  .   .  .  from  one  end  of  the  line,  the  most  probable 
a,     a, 

*  Harmonia  Mensurarum.  Cambridge,  1722.  (Quoted  by  Hultman,  Minsta  Qvadr. 
Stockholm,  1860.) 


DIRECT   OBSERVATION'S. 


103 


value  will  be  the  distance  X  of  the  centre  of  gravity  of 
these  weights  from  the  end  point,  and  will  be  found  by 
taking  moments  about  this  point  ;  that  is, 


and  therefore 


Ex.  To  test  the  power  of  the  telescope  of  the  great  theodolite  (3  ft.)  of  the 
English  Ordnance  Survey,  and  find  the  p.  e.  of  an  observation,  a  wooden 
framework  was  set  up  12,462  ft.  distant  from  the  theodolite  when  at  station 
Ben  More,  Scotland.  It  was  so  arranged  that  when  projected  against  the 
sky  a  fine  vertical  line  of  light,  the  breadth  of  which  was  regulated  by  the 
sliding  of  a  board,  was  shown  to  the  observer.  The  breadth  of  this  opening 
was  varied  by  half-inches  from  i^  in.  to  6  in.  during  the  observations;  which 
were  as  follows  :  * 


No.  of  obs. 

Width. 

Side  of  opening. 

Mean  of 
micr.  readings. 

I 

6.0 

(  left. 
<  right. 

(  28.00 
1  37oO 

2 

5-5 

11: 

(28.50 

1  37-t*. 

3 

5.0 

11: 

(  29.16 
(  37-16 

4 

4-5 

U: 

j  30.16 
f  36.66 

5 

4.0 

11: 

j  30.50 
<  37-16 

6 

3-5 

f  i. 

}r. 

j  31.16 

1  37-00 

7 

3-o 

11: 

(  32.66 
I  36.83 

8 

2-5 

il: 

<  33-50 
<  36.83 

9 

2.0 

jl: 

\  33-83 
/  37-00 

(  l. 

<  35-50 

10 

i-5 

<  r. 

1  37-16 

*  Account  of  the  Principal  Triangulation,  pp.  54,  55. 


104  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Let  X  =  the  most  probable  value  of  the  angle  subtending  an  opening  of 
i  inch.     Then  we  have  the  observation  equations 

6  X—  9.50  =  vi  3.5^—5.84  =  2/8 

5.5^—8.50  =  2/3  3  A"— 4.17  =  2/7 

5  X—  8.00  =  2/3  2.5  X—  3.33  =  z/8 

4.5  Af— 6.50  =  2/4  2.  X  —  3.17  =  2/8 

4  X '  —  6.66  =  2/6  1.5  X  —  1.66  =  z/10 

From   the   preceding  we  have  for  the  individual  values  of  A' and  their 
weights 

X  =  1.58     weight  62 
JT=  1.55     weight  5-52 


1.58X6'  +  1-55  X5-52  +   •   •    • 
.'.  weighted  mean  =  -       —  —  -  -  - 

=  1-55 
or  making  the  sum  of  the  squares  of  the  residuals  v  a  minimum,  that  is, 

(6A-  -  9.50)*  +  (5.  5  A"  -  8.  so)2  +  .  .  .  =  a  min. 
we  find  by  differentiation  that 

AT=i.55 
as  before. 

The  practical  rule  following  from  either  method  is  the  same,  and  may  be 
stated  thus  :  Multiply  each  observation  equation  by  the  coefficient  of  A"  in 
that  equation,  and  add  the  products.  The  resulting  equation  gives  the  value 
of  X. 

We  again  see  that  the  principle  of  minimum  squares  is  more  general  than 
that  of  the  arithmetic  mean,  and  why  it  was  that  we  failed  in  solving  the  equa- 
tions of  Art.  14.  In  other  words,  we  cannot  write  in  the  preceding  example 

X=  1.58 


and  take  the  mean  of  these  values  as  the  value  of  x,  because  these  equations 
should  be  written  strictly 


where  the  errors  are  not  lessened   in  the  same  proportion   throughout    the 
equations. 


DIRECT  OBSERVATIONS.  IO5 

Precision   of  a    Linear   Function    of  Independently    Observed 

Quantities. 

64.  Suppose  that  there  are  n  independently  observed 
quantities  J/,,  J/a,  .  .  .  whose  m.  s.  e.  are  //,,  /^,  ...  re- 
spectively, to  find  the  m.  s.  e.  /j.  of  F  where 

F=alMl  +  atMt+  .  .  .  +anAfn  (i) 

#„  av  .  .   .  an  being  constants. 

This  has  been  already  solved  in  Art.  19,  where  it  was 
shown  that 

2          r     2     an 

H  =  [a  [t.  ] 

On  account  of  the  great  importance  of  this  result  we  add 
another  method  of  deriving  it. 

If  J,,  Ja,  .  .  .  denote  the  errors  of  Mlt  M^  .  .  .  we  shall 
have  the  true  value  T  of  F  by  writing  ^/,  +  -/,,  J/.,-)-  J2,  .  .  . 
for  J/,,  M9,  ...  in  the  above  expression  for  F  ;  that  is, 


,  +  4)+-  .  .  +«.(-*/.  +40 

Call  J  the  error  of  F;  then,  since  T=  F-\-  J,  we  have 

J  =  ^1J1  +  ^.2J2+.    .    .  +anJH 
and  .*. 

ff  =  aW  +  af4+.  .  .+2^,44+-  •   • 

Let  the  number  of  sets  of  Mlt  M^,  .  .  .  required  to  find  T 
be  n,  and  suppose  J"  summed  for  all  the  sets  ol  values  of 
J,,  4,,  .  .  .  and  the  mean  taken,  then  attending  to  Art.  20, 


In  forming  all  possible  values  of  J,-/,,  J2JS,  .  .  .,  the 
number  of  values  being  very  large,  there  will  probably  be 
as  many  -j-  as  —  products  of  each  form,  and  we  therefore 
assume 

[J1JS]  =  [J1JI]=.  .  .  =o 
Hence 

^=[>y]  (3) 


Io6  THE  ADJUSTMENT   OP  OBSERVATIONS. 

Ex.  i.  The  Kevveenaw  Base  was  measured  with  two  measuring  tubes 
placed  end  to  end  in  succession.  Tube  i  was  placed  in  position  967  times, 
and  tube  2  966  times.  Given  the  p.  e.  of  the  length  of  tube  i  =  ±  o*«.ooo34, 
and  of  tube  2  =  ±  o'".ooo37,  find  the  p.  e.  in  the  length  of  the  line  arising 
from  the  uncertainties  in  the  length  of  the  tubes. 

[  p.  e.  from  tube  i  =  967  X  0.00034  =  o»*.329 
p.  e.  from  tube  2  =  966  X  0.00037  =  o"*.357 
.'.  p.  e.  of  line  =  Vo.^2(f  +  0.3572 


Ex,  2.  In  the  Keweenaw  Base  the  p.  e.  of  one  measurement  of  94  tubes, 
deduced  from  the  discrepancies  of  six  measurements  of  these  94  tubes,  was 
found  to  be  o'«.o3  .  Show  that  the  p.  e.  in  the  length  of  the  line  of  1933  tubes 
arising  from  the  same  causes  ma}'  be  estimated  at  ±  o*".i36. 

[  p.  e.  of  i  measurement  of  i  tube  =  —  '-^ 

1/94 

0.03 
p.  e.  of  base  of  1933  tubes  =  —  -± 


=  ±  0.136     J 

Attention  is  called  to  these  two  problems,  from  the  im- 
portance of  the  principles  illustrated.  In  Ex.  i  the  p.  e.  of 
a  tube  was  multiplied  by  the  whole  number  of  tubes  to  find 
the  p.  e.  of  the  base  from  that  cause,  for  the  reason  that 
with  whatever  error  the  tube  is  affected  it  is  cumulative 
throughout  the  measurement. 

In  Ex.  2  the  p.  e.  of  one  tube  is  multiplied  by  the  square 
root  of  the  number  of  tubes,  because  each  measurement  is 
independent  of  every  other,  and  the  errors  are  as  likely  to 
be  in  excess  as  in  defect,  and,  therefore,  may  be  expected  to 
destroy  one  another  in  the  final  result. 

Ex.  3.  The  m.  s.  e.  of  aM\,  /<  being  the  m.  s.  e.  of  M\  and  a  a  constant,  is 
equal  to  aju,  but  the  m.  s.  e.  of  the  sum  of  the  a  independently  observed 
quantities  Mi,  Mi  .  .  .  Ma  ;  that  is,  of  Mi  +  M*  +  .  .  .  +  Ma,  the  m.s.  e.  of 
each  being  ju,  is  Vaju.  Explain. 

65.  If  the  function  F  whose  m.  s.  e.  is  required  is  not 
in  the  linear  form,  we  first  reduce  it  to  that  form,  as  in  Art. 
7,  and  apply  Eq.  3,  Art.  64.  Thus,  if 


DIRECT   OBSERVATIONS. 


the  true   value  T  of  F  will   result    if  we    write 
M^-^-dM^,  .  .  .  for  J/p  J/,,  .  .   .  the  differentials  representing 
the  errors  of  these  quantities.     Then 


Expanding   by   Taylor's    theorem    and    retaining   only    the 
first  powers  of  the  small  quantities  ^/J/,,  dM.^  .  .  .  we  have 


T=f  + 


or 

Error  of  F— 

where 


This  expression  is  of  the  same  form  as  (i),  Art.  64.     Hence 

HP  =  a?ti?  -f  tf,V/,*  +  .  .  .  -f  a^i'-n  —  fay  ] 

The  still  more  general  case  of  the  m.  s.  e.  of  a  function  of 
quantities  which  are  themselves  functions  of  the  same  ob- 
served quantities  may  be  readily  reduced  to  the  form  of 
Eq.  i.  The  whole  point  is  to  express  the  error  of  F 
as  a  linear  function  of  the  errors  of  the  independently  ob- 
served quantities  J/,,  J/,  .  .  .  J\In. 

It  is  useful  to  note  that  Eq.  i  results  from  differ- 
entiating the  function  equation  directly,  as  has  been  al- 
ready pointed  out  in  Art.  /. 

Ex.  i.    If  /i,,  it.,  are   the   m.  s.  e.  of  the  measured  side?  AB,  BC  of  a  rect- 
angle ABCD,  find  the  m.  s.  e.  of  the  area  of  the  rectangle. 
[Here 

F-A^M-, 
.'.  by  differentiation 

,/F=Jlfl<Uf.1  +  J/,,/J/, 
and 

HP-  —  /I/,  -//..-    t   Af-i'nt'-\ 

15 


108  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  2.  The  expansions  of  the  steel  and  zinc  bars  of  tube  i  of  the  Repsold 
base  apparatus  of  the  U.  S.  Lake  Survey  for  i°  Fahr.  are  approximately 

5  =  0.0248  ±  o.oooi 
Z  '—  0.0617  ±  0.0003 

Show  that 

S      2        i 

—  -—-  ±  —    nearly. 

Z      5      400 

[For  F= 


and 

(p.  e.)-  =  —,  (0.0001)-  +  —  4  (o.ooos)2] 

Ex.  3/iThe  base  b  and  the  adjacent  angles  A,  C  of  a  triangle  ABC  are 
measured.  If  their  m.  s.  e.  are  respectively  //*,  J.IA,  Uc,  find  the  m.  s.  e.  of  the 
angle  B  and  of  the  side  a. 

To  find  /.IB- 

We  have 

y?  =  xSo  +  £  —  A  —  C 

where  £  denotes  the  spherical  excess  of  the  triangle. 
Hence,  A  and  C  being  independent  of  one  another, 

MB"-  HA*  +  l-ic'- 
To  find  jita. 

,  sin  A 

a  =  b-  -  - 

sin  B 

By  differentiation, 

sin  A   „       ,  sin  (C—  e)    . 

da  •=.  —    --db+l>  —  -  sin  I    dA  +  a  cot  B  sin  i    dC 

sin  B  sin*  B 

and  therefore 

sin2  A  !>•  sin2  (C  —  s)  sin2  i" 

"a"  =  -^"T-f>t">    +  ~  -T^-      -  I-1  *    +  a  cot  B  sm    :    /'C2 

sin^  B  sin4  B 

Ex.  4.  Given  the  base  b  and  the  angles  A,B  of  a  triangle  with  m.  s.  e. 
Mb,  HA,  MB  respectively,  to  find  the  m.  s.  e.  f.ia  of  the  side  a. 

We  have 

,  sin  A 
a  •=  b  -  (i) 

sin  B 

This  might  be  expanded  as  in  the  preceding  example,  but  more  conveniently 
as  follows  : 

Take  logarithms  of  both  members.     Then 

log  a  =  log  b  +  log  sin  A  —  log  sin  B  (2) 


DIRECT   OBSERVATIONS.  109 


(a)  By  differentiation. 


da  =  -  db  +  a  cot  A  sin  I  "dA  —u  cot  B  sin  I  "dB 

b 
Hence 

a1 
Ha"2  —  —  Hb*  +  o'2cof2,-/  sin^i"//,!7  +  <x8cotj.fl  sin2i>s2  (3) 

0" 

If,  as  is  usually  assumed  in  practice, 

MA  =  MB  =  H,  and  /m,  =  o 
then 


f.ta  =  a  sin  i"  f.i  V'cot'M   +  cotfB  (4) 

(b)  Using  log  differences  as  explained  in   Chapter  I.,  we  have  by  differen- 
tiating (2) 

Sa  da  =  db  db  +  f)AdA  -  t)B  dB  (5) 


where  fta,  di>  are  the  differences  corresponding  to  one  unit  for  the  numbers  a 
and  b  in  a  table  of  logarithms,  and  dA,  SB  are  the  differences  for  i"  for  the 
angles  A  and  B  in  a  table  of  log  sines.  Hence 


•'• 

The  two  equations  (3)  and  (6)  may  be  used  to  check  one  another. 

Ex.  5.  The  following  example  is  given  for  the  sake  of  showing  the  form 
of  solution  by  the  method  of  logarithmic  differences. 

In  the  triangulation  of  Lake  Superior  there  were  measured  in  the  triangle 
Middle,  Crebassa,  Traverse  Id.  (ABC) 

Z  A  =  57°  04'  51".  4      HA  =  o".30 
Z  B  —  67°  15'  39".  2      /IB  —  o".29 

The   side    Middle-Traverse   Id.  as   computed   from   the    Keweenaw    Base  is 
16894.9  yards.     Taking  fti,  =  0.05  yd.,  find  //<»  and  //c 


We  have 


,  sin  A 

a  =  b 

sin  B 


.• .  log  (a  +  da)  =  log  (b  +  dt>)  +  log  sin  (A  +  dA)  —  log  sin  (B  +  dB) 

Then   (see  Ex.  i,  Art.  7),   the   differences   being   expressed    in    units   of  the 
seventh  decimal  place, 
log  (b  +  db)  =  4.2277556  +  257  db 

log  sin  (A  +  dA)       =  9.9239892  +    14  dA 
colog  sin  (B  +  dB)  —.  0.0351398  —      9  dB 
.  • .  log  a  +  $a  da         =  4. 1868846  +  257  db  +  14  dA  —  9  dB 
and  283  da  —      257  db  +  14  dA  —  ^dB. 

Since  log  a  =  4.1868846,  and  283  is  the  difference  Sa  as  given  in  the 

table. 


IIO  THE   ADJUSTMENT   OF   OBSERVATIONS. 


Hence  ,.„    —  , 

=  0.0024 

and  Ha  =  0.05 

Also 


=  (?g)'  (.05)'  +  (-£-)'  (.30) 


=    V'(o.29)2  +  (0.30)- 
=  o".42 


Miscellaneous  Examples. 

Y        >•"• 
66.  Examples  of  Mean-Square  and   Probable    Error. 

.£>.  i.  If  in  a  theodolite  read  by  2  verniers  the  p.  e.  of  a  reading  (mean  of 
vernier  readings)  is  2",  show  that  if  it  is  read  by  3  verniers  the  p.  e.  of  a 
reading  will  be  a  little  over  i".5,  and  if  read  by  4  verniers  a  little  less  than  i".5. 

Ex.  2.  The  p.  e.  of  an  angle  of  a  triangle  is  r  ;  show  that  the  p.  e.  of  the 
triangle-error  is  ;-  \  3,  all  of  the  angles  being  equally  well  measured. 

[Error  =  180°  -  (A  +  B  +  Q.] 

Ex.  3.  The  expansion  of  a  bar  for  i°  C.  is  ga  ±  9/7  show  that  for  i°  Fahr. 
it  is  $a  ±  $r. 

Ex.  4.  The  length  of  a  measuring  bar  at  the  beginning  of  a  measurement 
was  a  ±  >'i.  After  x  measures  had  been  made  it  was  b  ±  ;-2.  Show  that  the 
length  of  the  «th  measure,  the  length  being  supposed  to  change  uniformly 
witli  the  distance  measured,  is 


.~^-^ 


a+        6- 


[For   if  da   is    the   error  of  a,  and   db  of  />,  then  the  error  of  a  -\  —  (b  —  a)  is 

(i  --  ]da  +    -  db.  and  the  above  p.  e.  follows. 
x)  x 

It   is   a   common  mistake  to  write  the  error  in  the  form  da  +  -  (db  —  da), 

/  «« 

,'^  and  hence  to  infer  that  the  p.  e.  is  •{/  r^  -\  —  -(r^  +  r2'2).  ] 

Ex.  5.   Prove  that  the  p.  e.  of  the  mean  of  two  observations  whose  dif- 
ference is  d  is  0.337  d,  and  the  p.  e.  of  each  observation  is  0.477  d. 


DIRECT   OBSERVATIONS.  Ill 

Ex.  (i.  The  line  Monadnock-Gunstock  (94469  m.)  was  computed  from  the 
Massachusetts  Base  (17326  w.)  through  the  intervening  triangulation.  The 
p.  e.  of  the  line  arising  from  the  triangulation  is  ±  om.3i7,  and  the  p.  e.  of 
the  base  is  0^.0358  ;  find  the  total  p.  e.  of  the  line. 


[p.e.  =  f/(^§  X  0.0358)*  +(0.317)-  =  ±  o».372] 

Ex.  7.  The  Minnesota  Point  Base  reduced  to  sea-level  is 

1325  X  15  ft.  bar  at  32    4-  n"».3i4  ±  o'«.42i 
and  15  ft.  bar  at  32    =  179'".  95438  ±  o»'».oooi2 

show  that  the  p.  e.  of  the  base  is  ±  o»«.45o. 


[p.  e.  =:  1(1325  X  0.00012)- +  (0.421)-  =  ±  o'«.45o.  We  multiply 
±  o'«. 00012  by  1325  :  uncertain  which  sign  it  is  ;  but  whichever  it  is,  it  is 
constant  all  the  way  through.] 

E.\\  S.  If  the  zenith  distance  £  of  a  star  is  observed  ;/i  times  at  upper 
culmination,  and  trfe  zenith  distance  T  of  the  same  star  is  observed  «2  times 
at  lower  culmination,  show  that  the  m.  s.  e.  of  the  latitude  of  the  place  of 
observation  is 


V-+- 

2   '     HI       n-z 


H  being  the  m.  s.  e.  of  a  single  observation. 

[Latitude  =  90    —  f  (?  +  ?')! 

Ex.  9.   Given  the  telegraphic  longitude  results, 

h.     m.     s.  s. 

Cambridge  west  of  Greenwich  =  4  44  30.99  ±  0.23 
Omaha  west  of  Cambridge  =  i  39  15.04  ±  0.06 
Springfield  east  of  Omaha  25  08.69  ±  o.  n 

show  that 

Springfield  west  of  Greenwich  =  5   58  37.34  ±  0.26 

[p.  e.  =:    t  .23-  +  .06-  +  .11-  =  0.26] 
E.\.  10.  Given  mass  of  earth  +  mass  of  moon  = 


305879  ±  2271 

and  mass  of  moon  =  —    —  mass  of  earth 

prove  mass  of  earth  —  — 


309635  ±  2299 

[For  (305879  ±  22-1)  X  ^  -  =  309635  ±  2299] 
c  i .  44 


112  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  ii.  In  measuring  an  angle  suppose 

r^  =  p.  e.  of  a  pointing  at  a  signal, 

r-i  =  p.  e.  of  a  reading  of  the  limb  of  the  instrument, 

e   •=.  error  of  graduation  of  the  arc  read  on, 

then,  assuming  that  these  result  from  the  only  sources  of  error  not  eliminated, 
show  if  the  limb  has  been  changed  m  times  and  n  readings  taken  in  each 
position,  that 

p.  e.  of  angle  =  ± 
[For  one  position  of  the  limb 


/2(?V  +  rj) 
--  +  c2 

as  the  error  of  graduation  remains  constant  throughout  each  set  of  //  readings]. 

Ex.  12.  The  distance  o—  imm  on  a  graduated  line-measure  is  read  with 
a  micrometer;  show  that  the  p.  e.  of  the  mean  of  two  results  is  equal  to  the 
p.  e.  of  a  single  reading. 

[For  distance  o  —  imm  , 

=  ^  |  (first  +  second  rdg.)  at  o  —  (first  +  second  rdg.)  at  imm  \ 
.•.(p.e.)s=i{4(p.e.)sofardg.}] 

Ex.  13.  In  the  comparison  of  a  mm.  space  on  two  standards  placed 
side  by  side  and  read  with  a  micrometer,  the  p.  e.  of  a  single  micrometer  read- 
ing being  a,  show  that  the  p.  e.  of  the  difference  of  the  results  of  M  combined 

/2 
measurements  (each  being  the  mean  of  two  measurements)  is  4/  -a. 

[For  p.  e.  of  a  reading  =  a 

p.  e.  of  a  combined  measurement  =  a 

and  p.  e.  of  mean  of  n  combined  measurements  =  —  —  ,  etc.] 

Vn 

Ex.  14.  A  theodolite  is  furnished  with  n  reading  microscopes,  all  of  the 
same  precision.  A  graduation-mark  on  the  limb  is  read  on  m  times  with  a 
single  microscope,  giving  the  p.  e.  of  a  single  reading  to  be  r±.  The  telescope 
is  then  pointed  at  an  object  m  times,  and  the  p.  e.  of  the  mean  of  the  micro- 
scope readings  is  found  to  be  r2.  Show  that  the  p.  e.  of  a  pointing  is 


[p.  e.  of  reading  (mean  of  verniers)  with  n  microscopes  =  — 

Vn 
Total  error  =  error  of  reading  +  error  of  pointing 

.'.  rj  =  —  +  (p.  e.  of  ptg.)5,  etc.] 
n 


DIRECT   OBSERVATIONS.  113 

Ex,  15.  If  //i£i,  n-ibi  are  the  m.  s.  e.  of  the  base  measurements,  and  fit\ 
the  m.  s.  e.  of  the  ratio  A,  given  by  the  triangulation,  of  a  base  b-t  to  a  base  b\, 
show  that  the  m.  s.  e.  of  the  discrepancy  between  the  computed  and  measured 
values  of  b*  is  bt  V[>«s]. 

[Discrepancy  =  £2  —  />]A  =  /  suppose 

db*-l>ld\—Xdl>l=dl 
and     fist**  +  /'is(;<3A)4  +  A-(/<,*,)2  =  n", 
or        /'./W+/<32 +  ,«,-)  =  /<-] 

Ex.  16.  At  the  time  t\  the  correction  to  a  chronometer  was  af  ±  rj,  and 
at  the  time  /2  it  was  a-f  ±  ray  show  that  the  p.  e.  of  the  rate  of  the  chronometer 

is    -  —  and   find   the  p.  e.  of  the  correction  to   the   chronometer  at   an 

/a  —  /i 

interpolated  time  /'. 

I",-.                               ai  —  a>    , 
Correction  =  a±  -\ (/  —  A)  at  time  /  , 

L  I*  —  ^i 


.*.  p.  e.  = •• — 

Ex.  17.  Given  x  cos  a  =  A  ±  n 

x  sin  a  =  /2  ±  ra 
find  p.  e.  of  .v  and  of  or. 


,       .//I'-vr-  +  /2  ->./- 

.'.  p.e.  of  JT  —  I/  —  —  --  r^  —  .     Similarly  p.  e.  of  n:  = 

M"    +   l-i~ 


£"JT.  iS.  Given  on  a  line  measure  the  p.  e.  of  a  distance  OA  measured 
from  0  to  be  r\,  and  of  OB,  also  measured  from  O,  to  be  rt  ;  find  the  p.e.  of 
OD  when  D  is  the  middle  point  of  AB. 

+  OB) 


.-.  r  =  i  i  TV'  +  TV 
If  /-,  =  r2  =  i**,  for  example,  then  r  —  o»*.S,  when  //  =  one  micron. 

It  may  at  first  sight  appear  paradoxical  that  the  p.  e.  of  the  computed 
quantity  may  be  smaller  than  the  p.  e.  of  the  measured.  It  is  evident,  how- 
ever, that  the  error  of  OD  is  one-half  the  sum  of  the  errors  of  OA  and  OB. 
If  the  signs  of  the  errors  are  alike  the  error  of  OD  is  never  greater  than  the 
larger  of  the  errors  ;  if  the  signs  are  different  it  is  always  less.J 


114  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  19.  Given  the  p.  e.  of  .r  to  be  r;  find  the  p.  e.  of  log  x. 

r  j  i  mod.    , 

a  log  x  =  —    _  dx 

L  ' x 

mod.     ~l 

.  .  p.  e.  log  x  = r 

a-         J 

Ex.  20.  'In  the  measurement  of  the  Massachusetts  base  line,  consisting  of 
2165  boxes,  the  p.  e.  of  a  box,  as  derived  from  comparisons  with  the  standard 
meter,  was  ±  o'«.ooooo55,  the  p.  e.  from  instability  of  microscopes  in  meas- 
uring a  box  was  om.oooi27,  and  the  p.  e.  of  the  base  from  temperature  cor- 
rections was  o"*.o332.  Show  that  the  p.  e.  of  the  base  arising  from  these 
independent  causes  combined  is  o'*.o358. 

[p.  e.  —   y  (2165  x  o^.oooooss)9  -\-  (o»".oooi27  V  2165)" -\-(om. 0332)- 
=  ±  o>«.035S] 

Ex.  21.  Given  the  length  of  the  Massachusetts  base  to  be  17326™. 3763 
±  om.O3$&;  show  that  the  corresponding  value  of  the  p.  e.  of  its  logarithm  is 
8.973  in  units  of  the  seventh  place  of  decimals. 

[  log  (b  ±  0.0358)  =  log  b  ±  — ~  (0.0358)      See  Art.  7. 

log  mod.     9.6377843 
log  0.0358  8.5538830 

8.1916673 
log  b  4.2387077 

0.0000008973  3.9529596  ] 

Ex.  22.  The  m.  s.  e.  of  the  log.  of  a  number  A^  in  units  of  the  seventh 
decimal  place  is  ±  10.6  ;  find  the  ratio  of  the  m.  s.  e.  to  the  number. 

[  log  (N  +  v)  =  log  N  +  ~ rr1  v 


/mod.Y    . 
)   » 


/'-log  (N  +  »)=  I 

mod. 
.  .  —TV-  /'  =  10.6  -5-  io7 

//  T       n 

and 


N        4iooooJ 

67.  Examplc§  of  Weighting. 

Ex.  i.  The  weights  of  the  independently  measured   angles  BAC,  CAD, 
DAE  are  3,  3,  i  respectively  ;  find  weight  of  the  sum-angle  BAE. 


ti        ill 
-  =-+-  +  -,   .'.    wt.  -0.6 
wt.      3      3      i  -J 


DIRECT   OBSERVATIONS.  115 

Ex.  2.  If  X  =  atxi  +  <i«x.2  +  .  .  .  +  anxn,  and  p\,  p-i,  .  .  .  pn  are  the  weights 
of  xi,  xi,  .  .  .  xn,  and  P  the  weight  of  X,  show  that 

-ur-!*    ^"'^.^  I 

,  ^    O.  .   '       \   ^     ol  -p 

r~  >  f/-  • 

.J!'  V  £*  '    -£"-«•.  3.  Prof.  Hall  found,  from  observations  of  the  satellites  of  Mars,  that 

from   Deimos,  Mass  of  Mars  =  ,    and    from    Phobos,  Mass  of 

3095313  ±  3435 

Mars  = the   mass   being   expressed   in   the   common    unit. 

3078456  ±  10104 

Show  that,  taking  the  weighted  mean,  we  have  approximately 

Mass  of  Mars  =  — 

3093500  ±  3295 

Ex.  4.  On  a  graduated  bar  the  space  o  —  im-is  measured  anci  found  to  be 
im.  with  a  weight  i,  and  the  space  o  —  2m  is  measured  and  found  to  be  -2m. 
with  a  weight  2;  required  the  value  of  the  space  im  —  2m  and  its  weight  P. 

[Space  im  —  2m  =  im.  It  makes  no  difference  what  the  weights  are  so 
far  as  the  value  of  the  space  is  concerned. 

To  find  P.     (im  —  2'"  )  =  (o—  2"'  )  —  (o  —  i'"  ) 


1113 

-  =  -  +  -  =  -  and  P= 

P      i      2      2 


2  -i 
= 

3  J 

E.\.  5.  Given  the  weight  of  .\  ~  =  f,  show  that 
weight  of  log  *  = 


y 

Ex.  6.   If  .v  =     and  the  weight  of  v  is/,  then 
weight  of  .V  =  c-  p 

Ex.  7.   Given  the   results   for  difference  of  longitude,    Washington    and 
Key  West, 

m.      s.  s. 

1873,  Dec.  24,  19  01.42  ±0.044 
Dec.  26,  1.37  ±    .037 
Dec.  30,  1.38  ±    .036 
Dec.  31,  1.45  ±    .036 

1874,  Jan.     9,  i.  60  ±    .046 
Jan.    10,  1.55  ±    .045 
Jan.    ii,  19  01.57  ±  0.047 

show  that 

m.      s.  t. 

weighted  mean  =  19  01.460  ±  0.016 

weighted  mean  of  first  four  nights  =n  19  01.404  ±  0.019 
weighted  mean  of  last  three  nights  =  19  01.573  ±  0.027 

and  from  the  last  two  results  check  the  first. 

10 


Il6  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  8.  In  the  trianguiation  connecting  the  Kent  Id.  Base,  Md.,  and  the 
Craney  Id.  Base,  Va.,  the  length  of  the  line  of  junction  computed  from 

w.  m.  / 

Kent  Id.  Base      =  26758.432  ±  0.38  ^ 

Craney  Id.  Base  =  26758.176  ±0.43 
Show  that 

m.  m.  .V   A, 

(1)  Discrepancy  of  computed  values    =          0.256  ±  0.57  ' 

(2)  Most  prob.  length  of  junction  line  =  26758.32    ±  0.028 

Ex.  g.  In  latitude  work  with  the  zenith  telescope,  if  n  north  stars  are 
combined  with  s  south  stars,  giving  ns  pairs,  to  find  the  weight  of  the  combina- 
tion, that  of  an  ordinary  pair,  one  north  and  one  south,  being  unity. 

[Let  //  =  the  m.  s.  e.  of  an  observation  of  one  north  star  or  of  one  south 
star. 

Then,  as  though  combining  the  mean  of  n  north  stars  with  the  mean  of 
s  south  stars,  the  wt./  of  the  combination  is 


2ns      ~[ 
.'.    i>  — 

f      n  +  s    . 


"The  combination  of  more  than  two  stars  gave  some  trouble.  In  one 
case  there  were  3  north  and  4  south,  which  would  give  12  pairs,  but  with  a 

3X4 
weight  of  2—7—-  only.     In  this  and  al!  similar  cases  I  treated  the  whole  com- 

3~r4 

bination  as  one  pair  ;  that  is,  I  inserted  in  the  blank  provided  the  half-sum  of 
the  mean  of  the  declinations  of  north  stars  ;md  of  the  mean  of  the  declinations 
of  south  stars,  and  gave  the  result  a  higher  weight.  This  is  the  only  logical 
method."  (Safford,  Report,  Chief  of  Engineers  U.  S.  A.,  1879,  p.  1987.) 

For  a  series  of  examples  by  Airy  on  the  weights  to  be  given  to  the 
separate  results  for  terrestrial  longitude  determined  by  the  observations  of 
transits  of  the  rnoon  and  fixed  stars,  see  Atem.  Roy.  Astron.  Soc.,  vol.  xix. 

Ex.  10.  If  a  close  zenith  star  is  observed  with  a  zenith  telescope  first  as  a 
north  star,  and  immediately  after  as  a  south  star,  show  that  the  weight  of  the 
resulting  latitude  is  less  than  that  found  from  observing  an  ordinary  pair. 

Ex.  ii.  In  the  trianguiation  of  Lake  Ontario  the  angle  Walworth-Palmyra- 
Sodus  was  measured  as  follows  : 

In  1875,  with  theodolite  P.  and  M.  No.  i, 

74°  25'  05". 429  ±  o".2g,  mean  of  16  results  ; 
In  1877,  with  theodolite  T.  and  S.  No.  3, 

74°  25'  O4".6ii  ±  o".22,  mean  of  24  results  ; 
required  the  most  probable  value  of  the  angle  and  its  probable  error. 


DIRECT   OBSERVATIONS.  117 

[With  first  theodolite  p.  e.  of  a  single  obs.        =  o".2()  I  16  =  i".i6 
Wiih  second  theodolite  p.  e.  of  a  single  obs.  =  o".22  ^'24  =  i".oS 

Let  a  single  result  with  the  first  theodolite  be  taken  as  unit  of  weight, 
then  mean  of  16  results  has  weight  16. 

Let  a  single  result  with  the  second  theodolite  have  a  weight/,  referred  to 
the  same  unit  as  the  first,  then  mean  of  24  results  has  weight  24  p.  The  value 
of/  is  found  from  the  relation 


i       Vi.oS/ 
Also 

r".429  X  16  4-  o".6n  X 


most  prob.  value  of  angle  =  74'  25'  04"  + 


16  +  24/ 
and 

weight  of  this  value  =  16  +  24  /  ] 

Note.  —  If,  instead  of  being  two  measurements  of  the  same  angle,  the  above 
were  the  measurements  of  two  angles  side  by  side,  then 

total  angle  =  148°  50'  io".O4o 

because,  no  matier  how  much  better  one  is  measured  than  the  other,  we  can 
do  nothing  but  take  the  sum  of  the  two  values.  The  weight  P  of  the  result 
would  be  found  from 


£.v.  12.  An  angle  is  measured  n  times  with  a  repeating  theodolite,  and 
also  n  times  with  a  non-repeating  theodolite,  the  precision  of  a  single  reading 
and  of  a  single  pointing  being  the  same  in  both  cases;  compare  the  weights  of 
the  results. 

[  //i,  /'a  the  m.  s.  e.  of  a  single  pointing  and  of  a  single  reading. 

With  a  non-repeating  theodolite  each  measurement  of  the  angle  contains 

(pointing  +  reading)  —  (pointing  4-  reading) 
.'.  (m.  s.  e.)-  of  one  measurement  =  2//,5  +  2//2- 

and  (m.  s.  e.)-  of  mean  of  ;;  measurements  =  —  (2//r  +  2//22). 

« 

With  a  repeating  theodolite  the  successive  measurements  of  the  angle  are 

(pointing  +  reading)  —  pointing 
pointing  —  pointing 

pointing  —  (pointing  4-  reading) 
.'.  (m.  s.  e.)-  of  «  times  the  angle  =  2«/<,2  +  2fti- 

and  (m.  s.  e.)-  of  the  angle  =  —(aw//,1  4-  2//7!) 


Il8  THE   ADJUSTMENT   OF    OBSERVATIONS. 

If,  then,  p\,pi  denote  the  weights  of  an  angle  resulting  from  n  reiterations  or 
from  n  repetitions,  ^  ,.  .,     •        . 


and  hence  it  would  seem  that  the  method  of  repetition  is  to  be  preferred  to 
the  method  of  reiteration.  This  advantage  is  so  much  less,  the  smaller  —  -^  is  ; 

that  is,  the  more  the  precision  of  the  circle  reading  increases  in  proportion  to 
the  precision  of  the  pointing. 

This  result  is  contradicted  by  experience  —  so  much  so  that  in  all  of  the 
leading  surveys  repeating  theodolites  are  no  longer  used  in  primary  work. 
Where,  then,  is  the  fault?  Is  the  theory  of  least  squares  false?  By  no  means. 
We  have  only  another  example  of  a  point  which  occurs  over  and  over  again, 
and  which  is  so  apt  to  be  overlooked.  (See  Arts.  53,  54.) 

The  result  obtained  is  true  on  the  hypothesis  that  only  accidental  errors 
enter.  We  have  assumed  a  perfect  instrument  .  But  the  instrument-maker 
cannot  give  what  the  geometer  demands.  From  various  mechanical  reasons 
the  systematic  error  in  a  repeating  theodolite  increases  with  the  number  of 
observations,  whereas  in  the  reiterating  theodolite  it  disappears.  This  sys- 
tematic error,  in  whatever  way  it  arises,  causes  the  trouble.  It  is  useless  to 
discuss  accidental  errors  until  it  is  out  of  the  way;  and  as  no  means  have  yet 
been  devised  of  getting  rid  of  it,  the  instrument  itself  has  been  abandoned. 

Cf.  Struve,  Arc  dtt  Meridien,  vol.  i.  ;  Louis  Cruls,  Discussion  stir  les 
Me'thodes  de  repetition  et  reiteration,  Gand,  1875  ;  Herschel,  Outlines  of  A  str.  n- 
omy,  Art.  188  ;  Coast  Survey  Report,  1876,  App.  20.] 

Ex.  13.  An  angle  is  measured  with  two  repeating  theodolites.  With  the 
first  are  made  n,  repetitions,  //i,  /</  being  the  m.  s.  e.  of  a  pointing  and  of  a 
reading  ;  with  the  second  are  made  n^  repetitions,  //2,  Hi'  being  the  m.  s.  e.  of 
a  reading  and  of  a  pointing.  Show  that  if  «i,  «2  are  large  the  weights  of  the 
results  are  '•as 


NOTE    I. 

ON   THE   WEIGHTING   OF   OBSERVATIONS. 

68.  When  the  sources  of  error  are  of  such  kinds  that,  so 
far  as  we  know,  they  cannot  be  separated,  the  m.  s.  e.  and 
consequent  weight  are  found  as  described  in  the  preceding 
sections.  The  weight  has  been  defined  as  a  number  repre- 
senting the  relative  goodness  of  an  observation,  and  as  a 


WEIGHTING   OF   OBSERVATIONS.  I  19 

number  inversely  proportional  to  the  square  of  the  m.  s.  e. 
of  an  observation.  If  in  a  series  of  observations  the  con- 
ditions required  for  the  determination  of  the  law  of  error 
could  be  strictly  fulfilled,  these  two  statements  would  lead 
to  the  same  result.  In  actual  cases,  however,  this  is  only 
approximately  true.  Thus  two  separate  determinations  of 
a  millimeter  space,  made  in  the  same  way,  gave 

n  M 

looo.  i  ±  0.40,  mean  of  20  readings. 

1000.3  ±  °-33>  mean  of  30  readings. 

To  find  the  weighted  mean  of  these  two  sets  of  measure- 
ments we  may  proceed  in  two  ways.  The  number'  of  re- 
sults in  the  first  measurement  is  20,  and  the  number  in  the 
second  is  30.  Hence,  taking  the  weights  proportional  to 
the  number  of  results,  the  mean  . 


l=  IOQ0.220 


20  +  30 

Again,  since  the  p.  e.  of  the  measurements  are  0.40  and  0.33, 

their  weights  are  as  —  to  —  ,  that  is,  as  1080  to  1600.  and 

40'        33' 

the  resulting   weighted  mean  is    1000.219,  agreeing,  within 
the  limits  of  the  p.  e.,  with  the  other  value. 

In  this  example  the  two  methods  agree  as  nearly  as 
could  be  expected  from  the  small  number  of  observations. 
But  it  is  not  always  so.  Some  "  run  of  luck,"  or  balancing 
of  error,  or  constant  conditions  might  have  made  the  ob- 
servations of  one  set  fall  very  closely  together,  in  which 
case  the  weight  as  found  from  the  p.  e.  would  have  been 
very  large,  while  varying  conditions  might  have  caused 
wide  ranges,  giving  a  small  weight.  A  great  deal,  there- 
fore, depends  on  the  judgment  of  the  computer  in  deciding 
what  weight  is  to  be  given,  it  being  constantly  kept  in  mind 
that  the  strict  formulas  which  are  correct  in  an  ideal  case 
must  not  be  pressed  too  far  in  practice.  Thus  in  the  second 
set  of  observations  above  the  first  three  results  were  999.8, 


120  THE   ADJUSTMENT   OF   OBSERVATIONS. 

999.8,  999.8.  The  p.  e.  computed  from  these  would  be  zero, 
and  the  consequent  weight  infinite.  But  no  one  will  doubt 
but  that  the  mean  of  the  30  results  is  more  reliable  than  the 
mean  of  these  three  results. 

69.  An  Approximate  Method  of  Weighting. — A 
long-continued  series  of  observations  will  show  the  kind  of 
work  an  instrument  is  capable  of  doing  under  favorable  con- 
ditions ;  and  if  work  is  done  only  when  the  conditions  are 
favorable,  the  p.  e.  derived  from  a  certain  number  of  results 
will  generally  fall  within  limits  that  can  be  assigned  d  priori. 
For  example,  with  the  Lake  Survey  primary  theodolites, 
which  read  to  single  seconds,  the  tenths  being  estimated, 
the  work  of  several  seasons  showed  that  the  p.  e.  of  the 
mean  of  from  16  to  20  results  of  the  value  of  a  horizontal 
angle,  each  result  being  the  mean  of  a  reading  with  telescope 
direct  and  of  a  reading  witJi  telescope  reverse,  need  not  be 
expected  to  be  greater  than  d '.3.  If,  therefore,  after  having 
measured  a  series  of  angles  in  a  triangulation  net  with  these 
instruments,  the  p.  e.  all  fell  within  ±  o".3,  it  was  considered 
sufficiently  accurate  to  assign  to  each  angle  the  same 
weight. 

The  objection  to  this  is  that  "an  instrument  which  has  a 
large  periodic  error  may,  if  properly  used,  give  as  good 
results  as1  if  it  had  none ;  but  the  discrepancies  between  its 
combined  results  for  an  angle  and  their  mean  may  be  large, 
thus  giving  an  apparently  large  probable  error  to  the  mean. 
Moreover,  a  given  number  of  results  over  short  lines,  or 
lines  over  which  the  distant  signals  are  habitually  steady 
when  seen  in  the  telescope,  will  give  a  resulting  value  for 
the  angle  of  much  greater  weight  than  the  same  number  of 
combined  results  between  two  stations  which  are  habitually 
unsteady."  * 

The  same  method  of  weighting  was  employed  by  the 
Northern  Boundary  Commission  in  their  latitude  work. 
"  The  standard  number  of  observations  [for  a  latitude  de- 
termination] was  finally  fixed  at  about  60,  it  being  found 

*  Professional  Papers  of  the  Corps  of  Engineers  U.  S.  A.,  No.  24,  p.  354. 


WEIGHTING   OF   OBSERVATIONS.  121 

that  with  the  32-in.  instrument  60  observations  would  give 
a  mean  result  of  which  the  p.  e.  would  be  about  4  feet." 
70.  Weighting  when  Constant  Error  is  Present.— 

The  preceding  leads  us  to  the  case  where  the  error  of  ob- 
servation can  be  separated  into  two  parts,  one  of  which  is 
due  to  accidental  causes  and  the  other  to  causes  which  are 
constant  throughout  the  observations.  The  total  error  e 
would,  therefore,  be  of  the  form 


This  case  has  been  discussed  already  in  general  terms  in 
Art.  53  in  explaining  the  well-known  fact  that  an  increase  in 
the  number  of  observations  with  a  given  instrument  does 
not  lead  to  a  corresponding  increase  of  accuracy  in  the 
result  obtained. 
Let 

//,  =  the  m.  s.  e.  of  the    observation   arising   from   the 

accidental  causes, 
^  =  the  error  peculiar  to  the  observation  arising  from 

the  constant  causes. 

Then  /./,,  ^,  being  independent,  the  total  m.  s.  e.  [j.  of  obser- 
vation may  be  assumed 

!'•  =  /V  +  /V 

If  //  observations    have    been    made  we  shall   have  for  the 
m.  s.  e.  fjLg  of  their  mean,  since  //,,  is  constant, 


It  is  evident  that  when  ;/  is  large  «./  becomes  the  important 
term,  and  that  in  any  case  the  value  of  //„  and  consequent 
weight  can  be  but  little  improved  by  increasing  the  num- 
ber of  observations. 

For  the  purpose  of  finding  the  value  of  the  m.  s.  e. 
arising  from  the  constant  sources  of  error  a  special  series  of 
observations  is  in  general  necessary.  After  this  series  has 

*  Kffort,  Su  >";•)•  <>/  ///<•  y,»-//i,-rn  l^uihiary,  p.  86. 


122  THE   ADJUSTMENT   OF   OBSERVATIONS. 

been  made  the  value  of  ^  found  from  it  can  be  applied  in 
the  determination  of  the  value  of  /^  in  any  other  series  made 
under  like  conditions. 

For  illustration  let  us  consider  a  latitude  determination 
with  the  zenith  telescope.  It  is  well  known  that  with  this 
instrument  a  latitude  result  found  from  two  observations  of 
a  single  star  either  north  or  south  of  the  zenith  is  inferior  to 
one  found  from  a  combination  of  a  north  and  a  south  star. 
This  arises,  not  from  any  difference  in  the  mode  of  observa- 
tion, but  from  the  errors  in  declination  as  given  in  the  star 
catalogue  being  cumulative  in  the  one  case  and  balancing 
in  the  long  run  in  the  other. 

The  zenith-distance,  £,  of  each  star  being  observed,  the 
half-difference  of  zenith-distances  for  each  pair  may  be  com- 
puted, and  each  of  these  computed  values  may  be  con- 
sidered an  observed  value.  The  values  of  the  declinations 
3  are  taken  from  a  catalogue  of  stars.  The  errors  of  3  are, 
therefore,  independent  of  those  of  £,  and  are  constant  for  the 
same  pair  of  stars.  The  latitude  <p  from  one  pair  is  given  by 


Let 

fa  =  the  m.  s.  e.  of  £(£'  —  £)  for  one  observation  of  one  pair, 

fjis  =  the  m.  s.  e.  of  %  (3'  -f-  d)  for  this  pair, 

/4j,—  the  m.  s.  e.  of  the  resulting  latitude  <p  from  one  pair, 

then  for  a  single  observation  of  this  pair 


and  for  n  observations  of  this  pair 


The  quantity  //£  will  be  found  from  repeated  observa- 
tions of  the  same  pair  of  stars,  as  the  error  in  declination 
will  not  influence  the  result.  A  better  value  will,  of  course, 
be  obtained  from  several  pairs  than  from  a  single  pair.  Let, 
then,  many  pairs  of  stars  be  observed  night  after  night  for 


WEIGHTING   OF   OBSERVATIONS. 


123 


a  considerable  period.  Collect  into  groups  the  latitudes 
resulting  from  the  observed  values  of  each  separate  pair. 
Let  ;/.,  n.2,  .  .  .  vm  be  the  number  of  results  in  the  several 
groups,  the  number  in  any  group  being  at  least  two.  Form 
the  residuals  for  each  group  and  compute  the  m.  s.  e.  in  the 
usual  way.  We  have: 


First  Pair. 

Second  Pair.                  .   .   . 

No.  of 

Night. 

Results. 

z> 

Results. 

1  .  .  . 

I 

f,' 

v,7 

f.' 

•'./ 

.     .    . 

2 

n 

z,i" 

?; 

,.  /' 

3 

f'" 

.. 

<p* 

7'12 

Means 

f, 

f, 

... 

Now,  assuming  that,  the  m.  s.  e.  of  observation  of  each  pair 
is  the  same, 


•n,  —  i 


If,  then,  n  is  the  total  number  of  results,  and  ;//  the  number 
of  groups,  by  adding  the  above  equations  there  results 


In  finding  //^  we  assume  that  though  errors  of  declina- 
tion are  constant  for  each  star,  still  for  a  latitude  found 
from  many  pairs  in  the  same  catalogue  these  errors  mav  be 


124  THE   ADJUSTMENT   OF   OBSERVATIONS. 

regarded  as  accidental.  Let,  then,  many  different  pairs  of 
stars  be  observed  on  each  of  n  nights  at  m'  places,  no  star 
being  observed  at  more  than  one  place.  Collect  the  means 
of  the  single  results  of  each  separate  pair  and  form'  the 
residuals  v'  for  each  place,  tajdng  the  differences  between 
these  means  considered  as  single  results  and  their  mean  for 
that  place.  Then,  reasoning  as  above,  the  m.  s.  e.  of  a 
latitude  resulting  from  n  observations  on  a  single  pair  of 
stars  is 


QV] 


where  n'  is  the  number  of  different  pairs  of  stars  observed 
and  m'  is  the  number  of  places  occupied. 
Now,  /2s  is  found  from 


and  is,  therefore,  known  for  the  star  catalogue   used.     This 
value  may  be  taken  in  future  work  in  finding  /^  from 


and  the  consequent  combining  weight  of  y  will  be  as 

i 

/V 

An  example  of  a  similar  kind  is  afforded  in  finding  the 
weights  of  the  angles  measured  with  a  theodolite  in  a  tri- 
angulation  where  more  rigid  values  are  required  than  would 
be  found  by  Art.  69.  The  actual  error  of  a  measured  value 
of  an  angle  arises  from  two  main  sources,  errors  of  gradua- 
tion and  errors  of  observation  The  former  are  constant 
for  different  parts  of  the  limb  read  on,  and  correspond  to 
the  declination  errors  above,  while  the  latter  are  incapable 
of  classification,  and  are,  therefore,  assumed  to  be  accidental. 
The  periodic  errors  of  graduation  are  supposed  to  have 


WEIGHTING   OF   OBSERVATIONS.  125 

been  eliminated  by  proper  shiftings  of  the  circle.     The  re- 
sultant m.s.  e.  IJL  of  a  single  measurement  is  found  from 


and  the  m.  s.  e.  /*„  of  the  mean  of  n  measurements  made  on 
the  same  part  of  the  limb  from 

,   !+? 

«-•=*•+=• 

where  //„//„  are  the  m.s.  e.  of  graduation  and  observation 
respectively.  The  method  of  treating  this  problem  is  quite 
similar  to  that  of  the  preceding;  //2  is  found  by  reading  the 
same  graduation-mark  on  the  limb  many  times,  and  n,  by 
reading  the  angle  between  two  fixed  signals  many  times, 
the  limb  being  changed  after  each  reading.  Thence  («1  is 
known  for  the  instrument  in  question,  and  the  combining 
weights  of  angles  measured  with  this  instrument  are  at  once 
found. 

The  foregoing  leads  to  another  important  practical  point 
in  the  measurement  of  angles.  If  the  weight  of  a  single 
observation  is  unity,  then  the  weight  of  the  mean  of  ;/  ob- 
servations made  with  the  limb  in  one  position  is 

/=<"'+•"•' 


/v  +  -%- 

Experience  has  shown  that  we  may  safely  assume 
and  therefore  it  follows  that 


Hence,  no  matter  how  many  observations  we  make  in  one 
position  of  the  limb,  we  never  reach  the  precision  of  the 
mean  of  two  observations  made  with  the  limb  in  different 
positions. 

It    might    fairly    be    inferred    that    the    limb    should    be 
shifted  after  each  single  reading  of  an  angle,  and  the  rea- 


126  THE   ADJUSTMENT   OF   OBSERVATIONS. 

sons  for  not  doing  so  are  to  guard  against  mistakes  in 
reading  and  to  eliminate  twist  of  station,  as  explained  else- 
where. (Art.  54.) 

71.  Assignment  of  Weight  Arbitrarily. — So  far  we 

have  deduced  the  combining  weights  from  the  observed 
values  themselves,  or  from  them  in  connection  with  a 
special  series  of  observations.  But  this  may  not  always  be 
the  best  way  of  finding  the  weights.  The  observations  may 
not  be  our  only  source  of  information,  and,  indeed,  not  the 
most  reliable  source.  If,  for  example,  some  phenomenon 
has  been  observed  by  many  persons  in  different  parts  of  the 
country,  and  the  observations  are  sent  to  one  place  for  com- 
parison and  reduction,  it  would  not  be  proper  for  the  com- 
puter to  deduce  a  weight  for  each  series  from  the  observa- 
tions themselves  independent  of  other  sources  of  information 
he  might  have.  Some  of  the  most  inexperienced  observers 
with  the  poorest  instruments  might  have  apparently  better 
results  than  the  most  experienced  with  good  instruments. 
In  such  a  case  the  computer  must  exercise  his  own  judg- 
ment in  classing  the  observations.  He  should  consider  the 
experience  of  the  observer,  his  previous  record  for  accurate 
work,  the  kind  of  instrument  used,  the  conditions,  and  the 
observer's  record  of  what  he  saw — whether  it  is  clear  and 
precise  or  hazy  in  its  statements.  An  arbitrary  scale  of 
weights  may  then  be  constructed,  and  to  each  set  of  obser- 
vations be  assigned  a  weight  from  this  scale  according  to 
the  computer's  estimate  of  its  value.  No  two  computers 
would  be  likely  to  assign  precisely  the  same  weights,  but  if 
done  by  one  of  experience  and  good  judgment  the  result 
obtained  from  weighting  in  this  way  will  undoubtedly  be  of 
more  value  than  that  found  by  the  strict  application  of  the 
formulas  of  least  squares. 

The  point  is  simply  this.  The  class  of  observations  con- 
sidered may  be  expected  to  contain  systematic  errors  which 
cannot  be  determined,  and  is  therefore  not  capable  of  being 
treated  by  the  method  of  least  squares.  As  we  have  no 
direct  means  of  eliminating  this  kind  of  error,  we  must  do  so 


WEIGHTING   OF   OBSERVATIONS.  1 27 

indirectly  as  best  we  can,  and  that  is  what  the  system  of 
weighting-  mentioned  seeks  to  accomplish. 

An  example  will  be  found  in  the  discussion  of  the  Tele- 
scopic Observations  of  the  Transit  of  Mercury,  May  5-6,  1878, 
Washington,  1879,  where,  of  109  observations  sent  in,  to  only 
18  was  the  hig-hest  weight  assigned.  Prof.  Eastman,  under 

o  o  o 

whose  direction  they  were  reduced,  says:  "...  Several 
instances  may  be  found  where  small  weight  is  given  to  ob- 
servations that  apparently  agree  well  with  those  to  which 
the  highest  weight  is  assigned,  but  in  most  cases  the  ob- 
server's remarks  indicate  the  uncertain  character  ol  the 
observation." 

72.  Combination  of  Good  and  Inferior  Work.— 
It  is  strictly  in  accordance  with  the  idea  of  weight  that  if 
we  have  two  results  of  very  different  degrees  of  accuracy, 
a  result  better  on  the  whole  than  either  ma)-  be  found 
by  combining  both  with  their  proper  weights.  But  the 
proper  weights  may  be  difficult  to  find.  On  this  account  it 
depends  on  circumstances  whether  it  is  advisable  to  reduce 
a  set  of  observations  poorly  made,  in  order  to  combine  them 
with  a  well-made  set.  If  the  quantity  isavailable  for  observ- 
ing again  it  might  not  cost  any  more  to  do  this  than  to  reduce 
the  poor  observations.  Even  if  it  did  the  result  would  be 
more  satisfactory.  The  committee  of  the  Royal  Society  of 
England  which  was  appointed  to  examine  Col.  Lambton's 
geodetic  work  in  India  reported  that  "  Col.  Lambton's 
surveys,  though  executed  with  the  greatest  care  and 
ability,  were  carried  on  under  serious  difficulties,  and  at  a 
time  when  instrumental  appliances  were  far  less  complete 
than  at  present.  There  is  no  doubt  that  at  the  present  time 
the  surveys  admit  of  being  improved  in  every  part.  The 
standards  of  length  are  better  ascertained  than  formerly, 
and  all  uncertainty  on  the  unit  of  measure  may  be  re- 
moved. The  base-measuring  apparatus  can  be  improved. 
The  instruments  for  horizontal  angles  used  by  Col.  Lamb- 
ton  were  inferior  to  those  now  in  use.  .  .  .  The  com- 
mittee express  the  strong  hope  that  the  whole  of  Col. 


128  THE  ADJUSTMENT   OF   OBSERVATIONS. 

Lambton's  survey  may  be  repeated  with  the  best  modern 
appliances."  * 

73.  The  Weight  a  Function  of  our  Knowledge.— 

—If  a  quantity  is  not  available  for  observing  again,  as,  for 
example,  some  transient  phenomenon,  all  of  the  material  on 
hand  must  be  used,  and  the  best  weights  possible  assigned 
to  the  separate  values  in  order  to  combine  them.  The 
point  is  that  where  systematic  or  constant  error  has  not  been 
eliminated  the  weight  to  be  assigned  is  a  function  of  the 
state  of  our  knowledge — is,  in  fact,  a  matter  of  individual 
judgment. 

This  is  brought  out  very  fully  in  the  methods  used  in 
combining  the  older  star  catalogues  with  the  more  modern 
ones.  Thus  Safford  (Catalogue  of  Mean  Declinations  of  2018 
Stars,  Washington,  1879)  says:  "In  computing  positions  I 
have  generally  employed  Argelander's  rule  giving  to  a 
modern  determination  from 

1  observation  a  weight  |, 

2  observations  a  weight  £, 

3  to  8  observations  a  weight  i, 

*•*  o 

9  or  more  observations  a  weight  li  or  2. 

Argelander  generally  gives  Piazzi  a  weight  equal  to  unity  ; 
the  value  \  is  much  nearer  the  truth  ;  in  general  he  assigns 
rather  a  larger  relative  weight  to  the  older  and  poorer  ob- 
servations than  they  deserve.  But  this  is  mostly  compen- 
sated for  by  the  number  of  determinations." 

The  weight  of  a  quantity  being  a  function  of  our  know- 
ledge may  have  assigned  to  it  a  certain  value  at  one  time 
and  another  value  at  another  time  when  our  knowledge  of 
it  has  increased.  Thus  in  the  Fond  du  Lac  (Wis.)  base, 
measured  in  1872  with  the  Bache-Wiirdemann  compensating 
apparatus,  a  portion  was  measured  seven  times.  The  re- 
sults differed  widely,  far  beyond  what  was  expected  with 
the  apparatus.  No  reason  could  be  assigned  at  the  time  for 
the  discordances.  At  this  stage,  then,  one  would  have  been 

*  G.  T.  Survey  of  India,  vol.  ii.  p.  70. 


WEIGHTING  OF  OBSERVATIONS.  129 

justified  in  assigning  a  small  weight  to  the  value  of  the 
base. 

The  Keweenaw  base  was  next  measured  with  the  same 
apparatus,  and  the  same  trouble  came  in.  Next  the  Sandy 
Creek  base  and  then  the  Buffalo  base  were  measured.  Dur- 
ing all  this  time  (four  years)  material  had  been  accumulating 
for  the  explanation  of  the  behavior  of  the  apparatus.  When 
the  law  of  its  behavior  was  discovered  it  was  found  that 
good  work  not  only  could  be  done  but  had  been  done 
with  it. 

Hence  the  systematic  error  being  got  rid  of,  one  would 
be  justified  in  increasing  the  weights  of  the  bases  measured 
with  this  apparatus  in  comparison  with  bases  measured  with 
an  apparatus  of  a*  different  kind.  Had  the  later  work  not 
been  done  the  Fond  du  Lac  base  would  still  have  had 
assigned  to  it  the  low  weight. 

Take  another  instance.  Sir  G.  B.  Airy,  in  1847,  sajs  °f 
the  Mason  and  Dixon  arc  (Encyc.  Metrop.,  p.  209)  :  "  The 
results  of  this  measure  must,  we  think,  be  received  as  equal 
in  authority  to  those  of  any  other  measure."  This  may  have 
been  true  when  written  ;  but  Mr.  Schott,  in  1877,  in  his 
note  on  the  determination  of  the  figure  of  the  earth  from 
American  sources,  says  of  this  same  arc  (U.  S.  C.  S.  Report, 
1877,  p.  95):  "  It  is,  therefore,  only  owing  to  the  increased 
perfection  of  instrumental  means  and  methods  that  we  now 
dismiss  from  further  consideration  the  first  measured  North 
American  arc,  which,  moreover,  is  now  superseded  by  the 
present  measures." 

As  a  third  illustration  we  may  consider  the  weights  to 
be  assigned  to  a  system  of  differences  of  longitudes  in  which 
the  connections  of  the  stations  occupied  are  interlaced  as  in 
a  triangulation  net,  and  the  whole  svstem  is  to  be  adjusted 
so  as  to  remove  existing  contradictions. 

If  the  longitude  work  has  been  carried  out  on  one  plan, 
with  instruments  and  observers  of  about  the  same  quality, 
tnen  the  m.  s.  e.  of  each  determination  mav  be  computed 
from  the  measures  of  the  separate  nights,  and  in  the  adjust- 


130  THE   ADJUSTMENT   OF   OBSERVATIONS. 

ment  the  weights  may  be  taken  inversely  as  the  squares  of 
these  m.  s.  e. 

But  if  this  has  not  been  done,  if  in  the  older  work  in- 
struments, observers,  and  methods  were  poorer  than  later 
and  the  two  have  to  be  combined  in  the  adjustment,  the 
computer  must  estimate  as  best  he  can  their  relative 
weights.  Thus  in  a  system  in  Germany,  France,  and  Aus- 
tria reduced  by  Dr.  Albrecht*  the  observations  were  made 
between  the  years  1863  and  1876.  The  methods  of  obser- 
vation had  been  much  improved  in  this  interval.  In  as- 
signing the  relative  weights  a  scale  of  weights  was  first 
formed  from  a  consideration  of  all  the  knowledge  on  hand, 
taking  the  march  of  improvement  from  year  to  year  into 
account,  and  the  separate  determinations,  placed  in  one  or 
other  of  these  classes.  Thus,  for  example, 

Weight  i—  No  change  of  observers  ;  few  observations;  non- 
adjustment  of  electric  current ; 

Weight  2 — No  change  of  observers  ;  usual  variety  of  obser- 
vations ;  non-adjustment  of  electric  current ; 

Weight  3 — Change  ot  observers;  usual  variety  of  observa- 
tions ;  non-adjustment  of  electric  current, 

and  so  on. 

Similarly  Dr.  Bruhns  in  Verhandlungen  der  europdisclien 
Gradmessung,  1880.  See  also  Coast  Survey  Report,  1880, 
Appendix  6. 

74.  General  Remarks. — -The  subject  of  the  weighting 
of  observations  is  confessedly  a  difficult  one.  In  general  it 
may  be  affirmed  that  the  less  experienced  a  computer  is 
the  more  closely  he  will  adhere  to  the  rigorous  formulas 
without  considering  whether  systematic  errors  enter  or  not. 
As  he  adds  to  his  experience  he  will  consider  outside  evi- 
dence as  well  as  the  evidence  afforded  by  the  observations 
themselves.  This  will  be  specially  true  if  he  has  any  prac- 
tical knowledge  of  how  observations  are  made.  Indeed, 
it  is  doubtful  if  a  computer  can  apply  the  principles  of 

*  Astronomische  Nachricliten,  2132. 


REJECTION  OF  OBSERVATIONS.  131 

least  squares  properly  unless  he  is  at  least  an  average  ob- 
server. 

Great  caution,  however,  is  necessary  in  assigning  weights, 
because  it  is  sometimes  possible  so  to  choose  them  as  to 
make  observations  tell  anything  desired.  They  should 
always  be  chosen  from  a  consideration  of  all  the  evidence 
on  hand,  and  may  be  changed  as  additional  evidence  is  pre- 
sented, so  that  a  result  is  never  final,  but  is  ever  open  for 
improvement.  The  original  records  of  the  observations 
and  the  methods  of  reduction  are  quite  as  desirable,  if  in- 
deed not  more  so,  than  the  results  deduced.  The  best  plan, 
therefore,  is  to  publish  all  of  the  data  along  with  the  reduc- 
tion, when  the  reader,  if  he  wishes,  can  make  a  reduction 
for  himself.  He  can  then  form  a  more  intelligent  opinion 
of  the  computer's  skill  and  judgment,  and  also  of  the  value 
of  the  work. 


NOTE  II. 

ON    THE    REJECTION   OF   OBSERVATIONS. 

75.  There  is  nothing  in  the  \vhole  theory  of  errors  more 
perplexing  than  the  question  of  what  shall  be  done  with  an 
observation  of  a  series  which  differs  widely  from  the  others. 
In  making  a  series  of  observations  the  observer  is  given  full 
power.  He  can  vary  the  arrangements,  choose  his  own 
time  for  working,  reject  any  result  or  set  of  results;  he  can 
do  anything,  in  fact,  that  in  his  best  judgment  will  tend  to 
give  the  best  value  of  the  observed  quantity.  But  when  he 
has  finished  observing  and  goes  to  computing,  has  he  the 
same  power?  Can  he  alter,  reject,  manipulate  in  such  a 
way  as  in  his  best  judgment  will  give  a  result  of  maximum 
probability  ?  As  observer  he  was  supreme  ;  as  computer  is 
he  supreme,  or  only  in  leading-strings?  Various  answers 

iS 


132  THE   ADJUSTMENT   OF   OBSERVATIONS. 

may  be  given  to   this  question,  as  we  look  at  it  Irom  one 
point  of  view  or  another. 

In  the  hypothetical  case  on  which  the  exponential  law 
of  error  was  founded  there  were  no  discontinuous  observa- 
tions taken  into  account.  There  we  contemplated  not  only 
observations  made  with  the  best  instruments  and  by  the 
most  experienced  observers,  but  observations  of  all  grades, 
from  this  highest  grade  down  to  those  made  with  the  poorest 
instruments  and  by  the  most  ignorant  and  careless  observers 
conceivable.  It  is  only  in  this  way  that  errors  continuous 
all  the  way  from  -f-  co  to  —  QQ  could  arise.  In  the  cases 
occurring  in  ordinary  work  we  confine  our  attention  to  one 
section  of  the  observations  only — that  made  with  the  good 
instrument  and  by  the  skilful  observer.  This,  to  "be  sure,  is 
the  most  important,  and,  as  shown  in  Art.  30,  the  result 
following  from  it  differs  ordinarily  but  little  from  that  found 
in  the  ideal  case.  But  we  are  naturally  confronted  with 
difficulty  when  we  try  to  deal  with  a  very  incomplete  series. 
Extra  assumptions  must  be  made,  and  it  is  not  to  be  won- 
dered at  that  no  solution  yet  offered  is  regarded  as  entirely 
satisfactory. 

76.  A  common  summary  method  of  disposing  of  the  sub- 
ject is  contained  in  the  following  statement:  "The  weights 
[of  the  angles]  would  have  been  materially  increased  in 
many  instances  by  rejecting  what  would  appear  bad  obser- 
vations;  but  the  rule  has  been  never  to  reject  any  unless  the 
observer  has  made  a  remark  to  the  effect  that  it  ought  to 
be  rejected."  This  statement,  however,  does  not  cover  the 
whole  ground.  Those  who  reason  in  this  way  make  a  dis- 
tinction between  mistakes  and  errors  of  observation.  Mis- 
takes are  rejected.  But  the  great  difficulty  is  to  tell  just 
where  mistakes  end  and  errors  begin. 

Given  a  set  of  measures  involving  discordances  unlocked 
for,  and  which  the  observer's  remarks  do  not  cover,  how 
shall  we  proceed?  Two  views  may  be  taken.  In  the  first 
place,  the  computer,  from  a  consideration  of  the  measures 
themselves  and  from  all  other  evidence  bearing  on  them 


REJECTION   OF   OBSERVATIONS.  133 

that  he  can  discover,  may  make  a  distinction  between  meas- 
ures and  mistakes  which  will  do  for  the  set  before  him. 
With  observations  of  another  kind  he  might  have  a  different 
mode  of  procedure.  Another  computer  might  have  differ- 
ent rules  altogether,  precisely  as  in  the  case  of  weighting 
as  explained  in  Arts.  71-74. 

To  put  the  discrepant  values  with  the  other  values,  and 
take  the  arithmetic  mean  of  all,  would  give  a  result  con- 
siderably different  from  what  would  be  found  by  omitting 
them.  It  would  take  a  great  many  good  observations  to 
balance  the  effect  of  a  single  widely  discrepant  one.  It  is, 
then,  for  the  computer  to  judge  whether  this  discrepant 
observation  shall  have  the  same  weight  as  the  others  or  a 
different  weight. 

It  may  happen,  indeed,  that  the  discrepancy  is  so  evi- 
dently a  "natural  mistake"  that  it  may  be  corrected  with- 
out a  doubt  from  the  evidence  furnished  by  the  other 
observations,  and  the  discrepant  observation  changed  so 
that  it  may  be  treated  as  a  good  one.  Thus  an  angle  may 
be  read  5'  or  10'  wrong,  or  a  micrometer  screw  may  be  read 
5  or  10  revolutions  out  of  the  way,  as  shown  by  the  rest  of 
the  observations,  and  the  like. 

Or  the  observations  may  be  arranged  in  well-defined 
groups,  and  if  the  computer  finds  that  he  cannot  account 
for  the  presence  of  unusual  discrepancies  in  a  certain  group, 
he  may  decide  to  reject  the  whole  group.  For  example,  in 
longitude  work  it  may  happen  that  one  of  the  time  stars  may 
give  a  clock  correction  differing,  say,  one  second  of  time 
from  that  given  by  fifteen  or  twentv  others  observed  on  the 
same  night.  Instead  of  rejecting  the  single  star  it  would 
perhaps  be  better  to  reject  the  whole  night's  work.  If 
necessary,  an  extra  set  of  observations  may  be  made  to  fill 
the  blank.  By  rejecting  a  group  no  hidden  law  can  be 
slighted,  for  if  any  exists  it  will  continue  to  reappear  in 
further  observations,  and  finally  to  reveal  itself. 

77.  In  the  second  place,  the  computer,  instead  of  trusting 
to  his  judgment,  may  call  in  the  aid  of  the  calculus  of  proba- 


134  THE   ADJUSTMENT   OF   OBSERVATIONS. 

bilities  and  seek  to  establish  a  test  or  criterion  for  the  rejec- 
tion of  observations  which  will  serve  for  all  kinds  of  obser- 
vations. Of  the  criterions  which  have  been  proposed  the 
earliest  is  due  to  Prof.  Peirce.  It  is  as  follows:  "Observa- 
tions should  be  rejected  when  the  probability  of  the  system 
of  errors  obtained  by  retaining  them  is  less  than  that  of  the 
system  of  errors  obtained  by  their  rejection  multiplied  by 
the  probability  of  making  so  many  and  no  more  abnormal 
observations."  A  proof  by  Dr.  Gould  will  be  found  in  the 
U.  S.  Coast  Survey  Report,  1854,  pp.  131,  132.  It  is  founded 
on  the  assumption  of  the  Gaussian  law  of  error. 

Another  criterion  "  for  the  rejection  of  one  doubtful  ob- 
servation "  is  given  by  Chauvenet  in  his  Astronomy,  vol.  ii. 
p.  565.  "  We  have  seen  that  the  function  [Art.  30] 


/f'~r     /,'A* 
'" 


represents  in  general  the  number  of  errors  less  than  a  which 
may  be  expected  to  occur  in  any  extended  series  of  obser- 
vations when  the  whole  number  of  observations  is  taken  as 
unity,  r  being  the  p.  e.  of  an  observation.  If  this  be  multi- 
plied by  the  number  of  observations  n,  we  shall  have  the 
actual  number  of  errors  less  than  a;  and  hence  the  quantity 

n-nO(f)  =  n\i-  8(t}\ 

expresses  the  number  of  errors  to  be  expected  greater  than 
the  limit  a.  But  if  this  quantity  is  less  than  \  it  will  follow 
that  an  error  of  the  magnitude  a  will  have  a  greater  proba- 
bility against  it  than  for  it,  and  may,  therefore,  be  rejected. 
The  limit  of  rejection  of  a  single  doubtful  observation  is,  there- 
fore, obtained  from  the  equation 


t  = 

2n 


REJECTION   OF   OBSERVATIONS.  13$ 

A  third  criterion  was  proposed  by  Mr.  Stone,  Radcliffe 
observerat  Oxford,  Eng.,  in  Month.  Not.  Roy.Astron.Soc.,  1868, 
1873,  in  these  terms  :  "  I  assume  that  a  particular  person,  with 
definite  instrumental  means  and  under  given  circumstances, 
is  likely  to  make,  on  an  average,  one  mistake  in  the  making 
and  registering  n  observations  of  a  given  class.  The  proba- 
bility, therefore,  is  that  any  record  of  his  of  this  class  of  ob- 
servations as  a  mistake  is  -n.  From  the  average  discord- 
ances among  the  registered  observations  of  this  class  we 
can  find  the  p.  e.  of  an  observation  in  the  usual  way,  and 
also  the  probability  of  an  error  greater  than  a  given  quan- 
tity, as  C.  Then  if  the  probability  in  favor  of  a  discordance 
as  large  as  C  is  less  than  that  of  a  mistake,  or^,  the  discord- 
ant observation  is  rejected." 

The  least  objectionable  criterion  based  on  mathematical 
principles  may,  I  think,  be  developed  from  the  principle 
laid  down  in  Art.  50,  where  the  maximum  error  was  esti- 
mated at  about  five  times  the  p.  e.  or  three  times  the  m.  s.  e. 
If,  therefore,  an  observation  differs  from  the  general  run  of 
the  series  by  more  than  this  amount  it  should  at  least  be 
bracketed  and  attention  be  called  to  it. 

78.  It  may  be  stated  that,  as  a  general  rule,  criterions  are 
apt  to  be  most  highly  esteemed  by  those  who  look  at  the 
observations  from  a  purely  mathematical  rather  than  from 
the  practical  observer's  point  of  view.  The  latter  is,  with- 
out doubt,  the  true  standpoint.  Every  observer  will,  con- 
sciously or  unconsciously,  construct  a  criterion  suited  to 
the  sort  of  work  he  is  engaged  in.  This  criterion  will  not 
necessarily  be  founded  altogether  on  mathematical  for- 
mulas. Indeed,  most  likely  it  will  not  be.  Nor  does  it- 
follow  that  the  criterion  adopted  in  any  special  series  is  of 
universal  application  or  will  receive  universal  assent.  In 
the  process  of  weighting  the  observer  will  not  assign 
weights  always  as  the  inverse  square  of  the  m.  s.  e.  It  is 
often  better  to  assign  them  arbitrarily  from  a  feeling 
founded  on  a  general  grasp  of  all  the  circumstances  con- 
nected with  the  making  of  the  observations.  In  like  man- 


136  THE   ADJUSTMENT   OF   OBSERVATIONS. 

ner,  and  on  this  same  feeling1,  he  will  found  his  criterion  for 
rejection.  There  is  no  uniform  rule  for  weighting-,  neither 
is  there  one  for  rejection. 

A  criterion  such  as  Stone's,  for  example,  would  be  very 
useful  in  the  work  for  which  it  was  proposed,  and  in  its 
proposer's  hands  would  be  of  much  value.  But  that  any 
one  without  the  insight  given  by  long  familiarity  into  the 
special  kind  of  work  from  which  this  criterion  arose  could 
apply  it  properly  is  not  to  be  expected.  As  already  pointed 
out,  in  the  case  of  weighting  the  only  thing  for  the  com- 
puter to  do  is  to  publish  all  of  the  observations,  including 
those  rejected,  along  with  his  reduction,  when,  if  more 
light  can  at  any  time  be  thrown  on  them,  a  new  reduction 
can  be  made.  Take,  for  example,  Bradley 's  observations  as 
reduced  by  Bessel  and  Auwers  (Art.  51).  It  cannot  be  too 
strongly  insisted  on  that  a  result  deduced  from  a  series  of 
observations  is  never  to  be  looked  on  as  final,  but  as  ever 
open  for  improvement. 

79.  The  difficulty  in  combining  single  observations  lies 
in  assigning  to  them  their  proper  weights.  We  have  as- 
sumed the  arithmetic  mean  to  give  the  most  probable  value. 
If  the  sources  of  error  could  be  separated,  so  that  to  each 
single  observation  could  be  ascribed  its  proper  weight,  the 
resulting  weighted  mean  would  be  nearer  the  truth  than 
the  direct  arithmetic  mean.  We  can,  therefore,  conceive  of 
a  better  value  than  the  arithmetic  mean  in  certain  cases.* 
This  has  been  already  pointed  out  in  Art.  n. 

We  may,  therefore,  consider  whether,  when  discrepant 
observations  occur  we  may  not  get  a  more  satisfactory 
result  by  ignoring  the  arithmetic  mean  altogether.  Sup- 
pose, for  example,  that  we  had  three  observed  values,  100, 
60,  61,  and  that  we  had  no  means  of  getting  any  further 
observations.  These  values  would  seem  to  show  that  the 
true  value  was  likely  to  be  nearer  60  than  100.  Just 

*  De  Morgan  (Encyc.  Metrop.,  "  Theory  of  Probability,"  p.  456)  suggested  that  the  combin- 
ing weights  might  be  found  from  the  observed  values  themselves,  but  he  did  not  develop  his  plan, 
and  it  is  apparently  fruitless. 


REJECTION    OF   OBSERVATIONS.  137 

how  much  nearer  is  the  question.  To  reject  the  observa- 
tion 100  would  be  without  reason,  and  to  take  the  arith- 
metic mean  of  all  three  would  ignore  the  evidence  afforded 
by  the  observations  themselves. 

The  observations  being  discontinuous,  the  question  is 
removed  from  the  theory  of  least  squares,  which  presup- 
poses continuity  (see  Art.  30),  and  must  be  treated  by  other 
methods.  Laplace*  discussed  this  problem  long  before 
Legendre  and  Gauss  developed  the  method  of  least  squares. 
The  subsequent  confusion  in  the  introduction  of  criterions 
has  arisen  from  trying  to  fasten  to  the  method  of  least 
squares  what  is  in  reality  a  very  different  question. 

When  the  observed  values  are  discontinuous  we  cannot 
reasonably  assume  the  value  of  the  unknown  sought  to  be  a 
symmetrical  function  of  them  (see  Art.  11).  A  plausible 
result  has  been  given  (Arts  11,  15)  as  that  observed  value 
which  has  as  many  observed  values  greater  than  it  as  it  has 
less  than  it.  Thus  in  the  preceding  example  a  good  value 
to  choose  would  be  61. 

In  discussing  such  problems,  so  long  as  not  much  greater 
plausibility  can  be  assigned  to  one  method  of  combination 
than  to  another,  the  question  of  convenience  of  computation 
comes  in.  In  respect  to  this  the  propriety  of  selecting  the 
middle  term  stands  pre-eminent. 

80.  It  is  ever  to  be  kept  in  mind  that  unexpected  dis- 
crepancies in  his  results  do  not  always  prove  to  the  observer 
that  his  work  is  bad,  any  more  than  a  close  agreement 
among  them  shows  it  to  be  good.  When  either  occurs 
great  caution  is  necessary.  It  is  unsafe  to  have  a  rigid  rule 
of  any  kind  for  sifting  the  observed  values,  not  allowing 
the  computer  to  make  use  of  evidence  outside  of  the  ob- 
servations. By  following  such  rules  we  are  apt  to  bar 
the  way  to  discovery  of  new  truths,  or  at  least  to  hinder 
progress  in  that  direction.  See,  for  example,  the  history 
of  the  discovery  of  personal  equation,  American  Cyclopicdia, 
vol.  xiii. 

*  M4in.  Acati.  Paris,  vol.  vi.  p.  634. 


138  THE  ADJUSTMENT   OF  OBSERVATIONS. 

Throughout  this  discussion  it  has  been  assumed  that  the 
observations  have  been  reduced  by  the  observer  himself  or 
by  a  computer  who  is  at  the  same  time  a  competent  ob- 
server. A  computer  who  is  not  an  observer  must  of  ne- 
cessity employ  the  same  criterion  always;  and  in  this  case 
the  criterion  derived  from  Art.  50,  as  stated  on  page  135,  is 
to  be  preferred. 

The  following  memoirs  may  be  consulted  in  addition  to 
those  already  mentioned  :  Gergonne,  Amiales  de  Math., 
vol.  xii.  pp.  181  seq.  ;  Peirce  in  Gould's  A  sir  on.  Jour., 
vol.  ii.  pp.  161  seq.;  Airy  in  do.,  vol.  iv.  pp  137,  138;  Win- 
lock  in  do.,  vol.  iv.  pp.  145-147  ;  Stone,  Month.  Not.  Roy. 
Astron.  Soc.,  vol.  xxviii.  pp.  165  seq.,  vol.  xxxiv.  pp.  9  seq.  ; 
Glaisher  in  do.,  vol.  xxxiii.  pp  391  seq.,  also  in  Mem.  Roy. 
Astron.  Soc.,  vol.  xxxix.  pp.  75  seq. 


CHAPTER  IV. 

ADJUSTMENT     OF     INDIRECT   OBSERVATIONS. 
Determination  of  tJie  Most  Probable  Values. 

81.  If  direct  measurements  of  a  quantity  have  been  made 
under  the  same  circumstances,  we  have  seen  that  the  arith- 
metic mean  of  these  measures  gives  the  most  probable  value 
of  the  quantity.  We  now  come  to  the  case  where  the 
quantity  measured  is  not  the  unknown  required,  but  is  a 
linear  function  of  one  or  more  unknowns  whose  values  are 
to  be  found.  This  is  the  more  general  form,  and  its  solution 
has  been  carried  a  certain  distance  in  Art.  14.  The  point 
stopped  at  was  the  combining  of  observations  of  different 
weights.  As  by  the  aid  of  the  law  of  error  this  can  now  be 
done,  we  proceed  to  finish  the  solution. 

Let,  as  in  Art.  14,  the  equations  connecting  a  series  of 
observed  quantities  J/j,  M.2  .  .  .  Mn,  n  in  number,  and  the 
independent  unknowns  X,  V,  .  .  .  ,  n{  in  number  (n  >  ;/,-),  be 

^r+/,,F+.   .  .  -L^M>  +  v, 

a,X+b,  F+  .   .   .  -  L,  =  J\L  +  »,  (i) 


where  #,.  /;,,  .  .  .  /,,,  .  .  .  Ln  are  constants  given  by  theory 
for  each  observation,  and  i\,  ?',,  ...?'„  are  the  residual 
errors  of  observation. 

In  practice  the  labor  of  handling  these  equations  will  be 
much  lessened  by  using  an  artifice  we  have  several  times 
already  employed  (see  Art.  41).  Let  A",  Y'  ,  ...  be  close 
approximations  to  the  value  of  A',  F,  .  .  .  found  by  ordinary 
elimination  from  a  sufficient  number  of  the  equations,  or  by 
some  other  method,  as  by  trial,  for  example,  and  put 

x-x^x,  r-  Y  =j>,... 
19 


140  THE   ADJUSTMENT   OF   OBSERVATIONS. 

where  ,r,  y,  .  .  .  are  the  corrections  required  to  reduce  the 
approximate  values  to  the  most  probable  values. 
Then  the  observation  equations  reduce  to 

a,x  -f  b,y  -f  .  .  .  —  /,  =  v, 

a,x  -\-  b,y  +  .  .  .  —  /„  =  v,  (2) 


where 


+  .  .  .  -Ln-Mn 

and  are,  therefore,  known  quantities. 

It  is  more  convenient  in  practice  to  omit  the  residuals 
and  write  the  observation  equations  in  the  form 

a^x  -\-b.y-\-  .  .  .  =/, 

"S  +  bty  4-  •  •  .  =  4  (3) 

<***  +  bny  -f  .  .  .  =  4 

always  keeping  in  mind  that  the  strict  form  is  as  in  (2). 
The  observation  equations  3  may  be  written 

a1x  =  /l' 

a^x  —  //  (4) 

anx  —  ln' 

if  the  values  of  y,  z,  .  .  .  are  supposed  to  be  known. 

Now,  it  has  been  shown  in  Art.  63  that  the  most  probable 
value  of  x  would  be  found  from  these  equations  by  taking 

I'  I  '  I  ' 

the   weighted    mean  of  the  separate  values  -L,  JL,  .  .  .  JL. 

a1  a^  an 

The  weights  of  these  values  are  in  the  same  section  shown 
to  be  as  a*,  a*,  .  .  .  an\     We  have,  therefore, 

_[aT\ 
- 


INDIRECT   OBSERVATIONS.  14! 

or,  by  putting  for  //,  //,  .  .  .  their  values, 

_  {al}  — \ab~\y  — \ac~\z  — .  .  . 

•* — p — =j 

[aa] 

Similarly   if  the  values  of  x,  z,  .    .    .   are   supposed    to   be 
known,  the  most  probable  value  of y  is  found  from 

_\bl-\-\ba\X -\bc-\z-.  .  . 

[W] 
and  so  on  for  ^,   .   .   . 

Hence  we  should  obtain  the  weighted  mean  values  of 
x,  y,  .  .  .  ,  that  is,  their  most  probable  values,  from  the 
simultaneous  solution  of  the  equations 


[aa] 
_  [£/]  -  \ba~\x  -  \bc~\z  -  ... 

~\bbT 

that  is,  from  the  simultaneous  solution  of  the  equations 

[>*>+[^]j,+  .  .  .  =  [*/] 

(5) 


which  equations  are  equal  in  number  to  the  number  of  un- 
knowns. They  are  called  normal  equations,  or,  better,  final 
equations.  We  have  thus  found  the  most  probable  values  of 
the  unknowns  in  a  series  of  observation  equations  by  taking 
the  mean.* 

*  Another  method  of  solving  a  series  of  observation  equations,  due  to  Richelot,  is  worthy  of 
notice. 

Take  the  simple  case  of  n  equations  involving  two  unknowns.  Let  the  equations  (»  >  2)  be 
written  in  the  strict  form 


to  find  the  values  of  .r,  y  which  satisfy  them  best. 

Multiply  the  equations  in  order  by  the  undetermined  factors  /(-,,  X-2,  ...£„,  and  add  ;  then  if 
^i,  £2,  .  .  .  kn  satisfy  the  condition  \,bk\  —  o,  we  have 

[*/]        [**] 
[*«]  ^  [*«] 

The  best  value  of  a-  must  be  that  in  which  the  second  member  is  as  small  as  possible,  and  this 


142  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  preceding  result  may  be  derived  in  a  manner  which 
will  perhaps  show  still  more  clearly  that  we  have  taken  the 
means  of  the  separate  values  of  the  unknowns 

For  simplicity  in  writing  consider  the  case  of  the  three 
observation  equations, 

a,x  +  bj  —  /, 
a,x  +  b,y  =  I, 
a*x  -f  b,  y  =  /3 

already  given  in  Art.  14. 

Solve  in  sets  of  two  in  all  possible  ways  by  the  method 
of  determinants,  and 

OA  -  a.b^x  =  bj,  -  bj, 
0/3  —  azb^x  =  bj,  —  bj, 
(a&  —  a.b,}x  =  bj,  -  bj, 

Take  the  weighted  mean  of  the  values  of  x  according  to 
Art.  63,  and  the  result  is 

_  (bill  —  bil^a^bi  —  a^i)  +  (b3l\  —  bil3)(aib3  —  a3i>i)  +  (b3Zi  —  b^l3}(a^b3  —  a3b^) 

' 


[aajbb]-  [abjab 
Similarly 

_[fl/][^]-MM 

\ad§bb~\-\ab\ab~\ 

happens  when  \kd\  is  as  great  as  possible.     But  \_ka\  as  a  linear  function  has  no  maximum  unless 
a  condition  exists  among  the  /t's  of  at  least  two  dimensions.     The  simplest  such  condition  is 

\kK\  =  i 
If  we  now  compute  the  maximum  value  of  [/(•«]  subject  to  the  conditions 

[bk]  -  o     [M]  =  i 
we  shall  find  as  the  best  value  of  x  (Todhunter,  Diff.  Calc.,  chap,  xvi.) 


\aa\\bb~\  -  \ab~\\aK\ 
which  agrees  with  the  value  resulting  from  the  normal  equations  5. 


INDIRECT   OBSERVATIONS.  143 

These  are  the  same  values  as  would  be  derived  from  the 
solution  of  the  equations 

\aa\x 
\bd\x 

Hence  if  in  a  series  of  n  observation  equations  containing 
i^  independent  unknowns  (it  >  wf)  -we  solve  all  possible  sets 
of  »,•  equations  by  the  method  of  determinants,  the  weighted 
means  of  the  separate  values  found  will  be  the  most  probable 
values  of  the  unknowns. 

This  method  of  solution  would  be  very  troublesome 
when  the  number  of  equations  is  large,  and  accordingly  we 
must  look  for  a  more  convenient  but  necessarily  equivalent 
method. 

Since  the  principle  that  the  sum  of  the  squares  of  the 
residual  errors  is  a  minimum  holds  whether  the  observed 
quantity  is  a  function  of  one  or  of  several  unknowns  (Art.  17), 
we  can  apply  it  to  the  simultaneous  solution  of  the  equations 
2.  The  residual  errors  must  satisfy  the  relation 

<',2  +  ''•/  +  •  •  -  +  v*  =  a  mill. 
that  is,  we  must  make 


+  .  .  .  +  aKX  +  bny  +  .  .  .  -  /„    =  a  mn. 

Now,  the  variables  x,y,  .  .  .  being  independent,  the  dif- 
ferential coefficients  of  the  expression  for  the  minimum  with 
respect  to  each  of  them  in  succession  must  be  equal  to  zero. 
Hence 


..-  =  o    (6) 


or,  collecting  the  coefficients  of  x,  y,  ...  in  each  equation, 

+  •  •  •  =[<! 

c\s  +  .  .  . 


144  THE  ADJUSTMENT   OF   OBSERVATIONS. 

The  number  of  these  equations  is  the  same  as  the  number  of 
unknowns ;  that  is,  nf.  Their  solution  will  give  the  most 
probable  values  of  x,  y,  .  .  .,  and,  adding  these  values  to  the 
approximate  values  X' ,  Y'  .  .'.  already  known,  the  most 
probable  values  of  X,  Y,  .  .  .  will  result. 

The  equations  7  are  identical  in  form  with  equations 
5,  found  by  taking  the  mean  of  all  possible  values  of  x  and 
y.  As  it  has  now  been  abundantly  shown  that  the  principle 
of  the  mean  and  that  of  minimum  squares  lead  to  the  same 
results  whatever  be  the  number  of  independent  unknowns, 
we  shall  use  whichever  promises  to  be  most  convenient  in 
any  particular  case. 

It  is  useful  to  notice  that  equations  6  may  be  written 

[av]  =  o 

r>]=0  (8) 

These  relations  correspond  to  \v\  =  o  in  the  case  of  the 
arithmetic  mean,  and  may  be  used  as  a  check  on  the  com- 
putation of  the  values  of  the  residuals. 

Ex.  Given  the  elevation  of  Ogden  above  the  ocean  by  C.  P.  R.  R.  levels 
to  be  4301  feet,  and  the  elevation  of  Cheyenne  to  be  6075  feet ;  also  the  ele- 
vation of  Cheyenne  above  Ogden  by  U.  P.  R.  R.  levels  to  be  1749  feet;  find 
the  adjusted  elevations  of  Ogden  and  Cheyenne  above  the  ocean,  supposing 
the  given  results  to  be  of  equal  value. 

First  solution  : 

(a)  Ogden — 

Direct  determination       4301     weight  i 
Indirect  determination   4326          "       \ 

.'.  weighted  mean       =  4309  feet. 

(b)  Cheyenne — 

Direct  determination       6075  weight     i 
Indirect  determination    6050        "         \ 

.'.  weighted  mean  =  6067  feet. 

Second  solution  : 

Let  X,  Y  denote  the  elevations  of  Ogden  and  Cheyenne  respectively. 
Then 

(^-  Y+ 


INDIRECT   OBSERVATIONS.  145 

Differentiate  with  respect  to  X,  Fin  succession,  and 

•2X-  7=2552 

-X  +  2^=7824 

.'  .  X  —  4309  feet. 

Y=  6067  feet. 

82.  If  the  observation  equations  are  of"  different  weights 
Pv  A»  •  •  •  A»  then,  reducing  each  equation  to  the  same  unit 
of  weight  by  multiplying  it  by  the  square  root  of  its  weight, 
we  have 


V  A 


with 

[/t'f]  ^=a  min. 

Substituting  the  values  of  \/pl  ?\,  Vp»vn_,  ...    in   the    mini 
mum  equation,  and  differentiating  with  respect  to  x,  y,  .  . 
as  independent  variables,  we  have  the  normal  equations 


...--=  [pbl]  (2) 


from  which  ,r,  y,  .  .  .  may  be  found. 

The  relations  for  weighted  equations  corresponding  to 
those  of  Eq.  6,  Art.  Si,  are  evidently 

\par~]  =o,  [X>''l=o,  ...  (3) 


Formation  of  the  Normal  Equations. 

83.  Instead  of  forming  the  minimum  equation  and  dif- 
ferentiating with  respect  to  the  unknowns  in  succession,  it 
is  more  convenient  to  proceed  according  to  the  following 
plans  suggested  by  the  form  of  the  normal  equations  them- 
selves. 


146  THE  ADJUSTMENT  OF   OBSERVATIONS. 

The  first,  from  equations  6,  Art.  81,  may  be  stated  as  fol- 
lows :  To  form  the  normal  equation  in  x  multiply  each  obser- 
vation equation  by  the  coefficient  of  x  in  that  equation,  and 
add  the  results.  To  form  the  normal  equation  in  y  multiply 
each  observation  equation  by  the  coefficient  of  y  in  that 
equation,  and  add  the  results.  Similarly  for  the  remaining 
unknowns. 

The  second  is  suggested  by  the  complete  form  of  the  nor- 
mal equations  as  given  in  equations  7,  Art.  81.  Accord- 
ing to  this  plan  we  compute  the  quantities  [aai],  [ab],  .  .  .  [«/], 
etc.,  separately,  and  write  in  their  proper  places  in  the 
equations. 

The  equality  of  the  coefficients  of  the  normal  equations 
in  the  horizontal  and  vertical  rows  leads  to  a  considerable 
shortening  of  the  numerical  work  in  computing  these  quan- 
tities. Thus  with  three  unknowns,  x,  y,  z,  all  the  unlike 
coefficients  are  contained  in 

4-  \_ad\x  -f-  \_ab~\y  -f~  \_ac\z  =  [«/] 

>==[*/] 


Instead,  therefore,  of  computing  12  quantities,  only  9  are 
necessary,  as  the  remaining  3  can  be  at  once  written  down. 
With  n  unknowns  the  saving  of  computation  amounts  to 

1+2+3+  -    -    .+(*-  !)  =  *«(*-  I) 

quantities. 

If  the  observation  equations  are  of  different  weights  the 
formation  of  the  normal  equations  may  be  carried  out  pre- 
cisely in  the  same  way  as  in  the  preceding  as  soon  as  the 
observation  equations  have  been  reduced  to  the  same  unit 
of  weight. 

The  form  of  the  weighted  normal  equations,  however, 
shows  that  it  is  not  necessary,  in  order  to  obtain  the  coef- 
ficients [paa],  \_pab~],  ...  to  multiply  the  observation  equa- 
tions by  the  square  roots  of  their  weights,  and  form  the  aux- 


INDIRECT   OBSERVATIONS.  147 

iliary  equations  I,  Art.  82,  since  the  products  aa,  ab,  .  .  . 
multiplied  by  the  weights  of  the  respective  equations  from 
which  they  are  formed  and  summed,  give  [paa\,  [pab\,  .  .  . 
directly.  This  is  important  because  labor-saving. 

Ex.  i.  Given  the-observation  (or  error)  equations,  all  of  equal  weight, 

x  =  i 

x  +  y        =  3 

x  —  y  +  z  =  2 

—x—y+z=l 

show  that  the  normal  equations  are 

4-*'  +  y          =5 


Ex.  2.  The  expansions  .Vi,  x-2,  ,v3,  x4  for  i°  Fahr.  of  four  standards  of  length 
were  found  by  special  experiment  to  be  connected  by  the  following  relations 
at  a  temperature  of  62rj  Fahr.  (//  —  one  micron.) 

iJ- 

+  -Vi                                               =      39-945     weight  i 

-l-viv.                                    =        5.932  "  1  6 

+  -v3                       5-371  "  4 

+    -r2  —  1.0937^3             =        0.006  "  8 

+  4-va                           --v,  =  -    1.335  "  3 

+  a,                                        —  .v4=  +  14.833  "  6 

find  their  most  probable  values. 
[The  normal  equations  are 

+  7.r,                                                  -  6.r4  =  +  128.943 

+  72.000  A-O  —    8.750^-3  —  I2.r4  =  +     78.940 

-    8.750^2  +  13.569.1-3  =  +     21.432 

+  (_).r4  =  —    84.993 


and  x,  =  39.913,   .r2  =  5-932,  -v3  =  5-4°5>   -V4  =  25.075] 

An  example  will  now  be  given  to  illustrate  the  method 
of  forming  a  series  of  observation  equations  : 

E.\\  3.   At  Washington   the   meridian   transits  of  the  following  stars  were 
observed  to  determine  the  correction  and   rate   of  sidereal  clock  Kessels  No. 
132^1,  April  12,  1870.  at  11  hours  clock  time. 
20 


148 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


Star. 

Observed 
clock  time  of  transit,  T. 

Right  ascension 
of  star,  a. 

r  Leonis. 

h.      m.          s. 
II      21       17.98 

s. 

16.00 

v  Leonis. 

II      30      20.41 

18.51 

ft  Leonis. 

II      42      28.57 

26.57 

o  Virginis. 

II      58      38.15 

36.20 

?/  Virginis. 

12       13       18.37 

16-37 

6  Virginis. 

13       3     16.36 

14-39 

Let  x  =  corr.  of  clock  at  n  hours  clock  time, 
y  =  rate  per  hour  of  clock. 

Now,  from  theoretical  considerations*  it  is  known  that  the  equation 
x  +  y(T—  n)  =  a  —  T 

gives  the  relation  between  the  clock  correction  and  rate  and  the  clock  time  of 
transit  of  each  star  observed. 

Hence  the  observation  equations  are 

^•  +  0.35^=  —  1.98 
x  +  0.507=  —  1.90 

X  +  O.7I_J/=  —  2.OO 
*  +  0.987=  —  1.95 
X  +  1.22}'=  —  2.OO 

x  +  2.057=  —  !-97 
For  the  remainder  of  the  solution  see  Art.  104. 

Ex.  4.  In   the  triangulation   of  Lake   Superior   executed   by   the   U.   S. 
Engineers  there  were  measured  at  station  Sawteeth  East  the  angles 

Farquhar-Porcupine  62°  59'  40". 33  weight  5 

Farquhar-Outer  64°  n'  34". 92  "       7 

Farquhar-Bayfield  100°  20'  29".  12  "      4 

Porcupine-Bayfield        37°  20'  49". 55  "      7 

Outer-Bayfield  36°  08'  55". 86  "      4 

required  the  adjusted  values  of  the  angles. 

All   of  the   angles  may  evidently  be   expressed    in 
terms  of  FSP,  FSO,  FSB.     Let  X,  Y,  Z  denote  the  most 
probable  values  of  these  angle?,  and   let  X',   Y',  Z'  be 
assumed    approximate   values   of  these   most    probable 
values,  and  x,y,  z  their  most  probable  corrections.     Denoting   the   measured 


Fig.9 


*  See  Chauvenet,  Astronomy,  vol.  ii.  chap.  v. 


INDIRECT    OBSERVATIONS.  149 

angles  in  order  by  M\,  M^  .  .  .  M-,,  and  their  most  probable  corrections  by 
Vi,  v-i,  ...  ^5,  we  have 

X'  +jc=      X         =  Mi  +  vi 

Y'  +y  =       Y         =  J/2  +  v* 

Z'  +  Z=  Z  =  M3  +  V3 

—  X'  —  x+Z'  +  z  =  —X  +  Z=  M<  +  vt 
-  V  -y  +  Z'  +  Z--V  +  Z=M6  +  v* 

For  simplicity  the  assumed  approximate  values  may  be  taken  equal  to  the 
observed  values  of  the  angles,  so  that  we  have  the  reduced  observation 
equations 

+  x  =  z/i     weight  5 

+  y  =v*         "7 

+  z  =v*  "4 

—  x  +  2  —  0.76  =  7/4  "      7 

— y   +  z— 1.66  =  7'5  "      4 

Hence  the  normal  equations 

izx  -    72—  —   5.32 

+  iiy  —    42=—  6.64 

—  7-r—    47  +  152=  +11.96 

Solving  these  equations,  we  find 

x=—  o".os,  y  =  —  o".36  2=+o".68 

Hence  z>i  =  —  o".O5,  t/2  =  —  o".36,  ^3=+o".6S,  z'4  =  — o".O3,  vb=  —  o".62, 
and  the  adjusted  values  of  the  angles  are 

62°  59'  40".  28 

64°  n'  34". 56 

1 00°  20'  29 ".So 

37°  20'  49". 52 

36°  08'  55". 24 

We  might  have  used  z/i,  c'2,  .  .  .  v-,  for  the  corrections  without  introducing  the 
symbols  x,  y,  z  at  all. 

Ex.  5.   If  the  unknown  x  occurs  in  each  of  the  n  observation  equations 

—  x  +  b\y  +  f i  z  +   .   .   .    =  A     weight  i 

—  x  +  l>iy  +  c?  z  +    .   .   .   =  l-i  "      i 

these  equations  are  equivalent  to  the  reduced  observation  equations 

bly  +  dz-\-    ...=/,     weight         i 
k->y  4-  Ci  z  +    .   .   .    =  /2  i 


150  THE   ADJUSTMENT    OF   OBSERVATIONS. 

[For  the  normal  equations  found  from  the  first  set  after  eliminating  x  are  the 
same  as  the  normal  equations  formed  from  the  second  set  directly.] 

Ex.  6.   Instead  of  the  observation  equation 

ax  +  by  +  cz  +   .   .   .   =  I    weight  / 
we  may  write 

qax  +  qby  +...=£/    weigh  t  ^ 

84.  Control  of  the  Formation  of  the  Normal 
Equations.  —  A  very  convenient  check  or  control  is  the 
following.  Add  as  an  extra  term  to  each  observation  equa- 
tion the  sum  of  the  coefficients  of  x,  y,  .  .  .  and  of  the  abso- 
lute term  /  in  that  equation,  and  treat  these  added  terms  just 
as  we  do  the  absolute  terms.  Thus  let  j,,  sa_,  .  .  .  sn  denote 
the  sums,  so  that 

«!  +  b,  +  c,  +  .  .  .  +  /,  =  s, 


Multiply   each   of  these  expressions  by  its  a   and  add  the 
products,  each  by  its  b  and  add,  and  so  on  ;  then 


.  .  .+[/?]=[&] 

If  these  equations  are  satisfied  the  normal  equations  are 
correct.  Thus  each  normal  equation  is  tested  as  soon  as  it 
is  formed. 

Since  \aa\,  \ab\,  .  .  .  \al\  have  been  computed  in  forming 
the  normal  equations,  the  only  new  terms  to  be  computed 
in  applying  the  check  are  [as],  \_bs\,  .  .  .  \ls],  [//]. 

Various  modifications  may  readily  be  applied  to  suit 
individual  tastes.  Thus  the  absolute  term  may  be  placed 


INDIRECT  OBSERVATIONS.  151 

on  the  other  side  of  the  sign  of  equality  ;  or  the  sign  of  the 
check  may  be  changed  so  as  to  make  the  sum  of  each  hori- 
zontal row  equal  to  zero. 

85.  Forms  of  Computing:  the  Normal  Equations.— 

When  the  number  of  unknowns  in  the  observation  equa- 
tions is  large,  or  when  their  coefficients  contain  several 
figures,  it  is  convenient  to  have  a  fixed  form  for  the  compu- 
tation of  the  terms  of  the  normal  equations.  It  lightens  the 
labor  much  either  in  forming,  solving,  or  in  finding  the  pre- 
cision of  the  unknowns  from  these  equations,  if  the  computa- 
tion is  so  arranged  that  a  check  can  at  all  times  be  applied 
and  the  whole  process  proceed  in  a  uniform  and  mechanical 
manner. 

The  aids  in  the  arithmetical  work  are  a  table  of  squares, 
a  table  of  products,  a  table  of  reciprocals,  a  table  of  log- 
arithms, and  an  arithmometer,  or  machine  for  performing 
multiplications  and  divisions.  The  latter  is  of  the  greatest 
use  in  computations  of  this  kind.  With  it  the  drudgery  of 
computation  is  in  great  measure  got  rid  of.  On  the  Lake 
Survey  two  forms  of  machine  were  used,  the  Grant  and  the 
Thomas.  A  series  of  trials  showed  that  with  either  ma- 
chine multiplications  could  be  performed  in  from  one-half 
to  one-third  of  the  time  required  with  a  log.  table,  and  with 
much  less  liability  to  error.  About  as  good  speed  can  be 
made  tor  a  short  time  with  Crelle's  multiplication-tables,  but 
it  cannot  be  kept  up. 

Form  (a).  With  Crelle's  tables,  or  with  a  machine,  the 
products  aa,  ab,  .  .  .  are  found  directly,  and  all  that  is  then 
to  be  done  is  to  \\rite  them  in  columns  and  take  their  sums 
|<7<?|,  [tf/>l,  .  .  .  With  a  Thomas  machine,  however,  each 
product  mav  be  added  to  all  that  precede,  so  that  the  final 
values  result  at  once. 

Let  us,  for  example,  take  the  observation  equations 

-  i .  2.1-  -|-  o.  2y  -j-  0.9  =n  7-, 
+  3.0*— 2.n«-f-  1.1=1', 


152 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


Arrange  as  follows,  the  headings  indicating  the  nature  of 
the  numbers  underneath  : 


a 

b 

/ 

J 

—  1.2 

+  O.2 

+  O.Q 

—  O.I 

+  3-0 

—  2.1 

+  I.I 

+  2.0 

+  0.7 

+  1.6 

-4.0 

-i-7 

act 

all 

al 

as 

1.44 
9.00 
0.49 

—  0.24 
—  6.30 

+  I.  12 

—  1.  08 
+  3-30 
-2.80 

+  O.  12 

+  6.00 
-1.19 

+  10  93 

-5-42 

—  0.58 

+  4-93 

bb 

bl 

bs 

-    5-42 

0.04 
4.41 
2.56 

+  0.18 

—  2.31 
—  6.40 

—  O.O2 
—  4.  2O 
-2.72 

+  7.01 

-8.53 

-6.94 

11 

Is 

-  0.58 

-8-53 

0.81 

I.  21 

16.00 

—  0.09 

+  2.2O 
+  6.80 

+  18.02 

+  8.9I 

Hence  the  normal  equations,  with  the  check  all  ready  for 
solution,  are 

<?=-+ 1 0.93.*-— 5.427-0.58  +4-93 

o-         5.42^+7.017-8.53  -6.94 

Form  (b).   If  logarithms  alone  are  used,  form  a  table  of 
the  log.  coefficients  of  the  observation  equations  as  follows : 

log  alt  log  £lt  .  .  .  log  /t,  log  s, 
log  «„  log  &,,...  log  /„  log  s, 

log  ant  log  bnt .  .   .  log  /„,  log  sn 

Write  log  al  on  a  slip  of  paper  and  carry  it  along  the  top 
row,  forming  the  products 

log  a,alt  log  afa  .   .   .,  log  «,/„  log  a,s, 


INDIRECT  OBSERVATIONS. 


153 


Similarly   with  log  a3  form   the  products  from  the  second 
row, 

log  a,a,,  log  aj)v  .   .   .,  log  ajv  log  a^ 

and  so  on  till  log  an  is  reached. 

The  numbers  corresponding  to  these  logarithms  are  next 
found,  so  that  we  have 


By  addition  we  find 

\aa],  [ab~\,  .  .  .  [>/],  DW] 

the  coefficients  of  the  unknowns  in  the  first  normal  equation. 
Proceed  in  a  precisely  similar  way  with  log  b^  log  />„ 
.  .  .  log  bH,  omitting  the  term  [ab~\  already  found;  with 
log  clt  log  cv  .  .  .  log  cn,  omitting  the  terms  \_ac\  [be]  already 
found  ;  and  so  on  till  the  last  quantity  is  reached, 


logo 

log  b 

log/ 

log  j 

0.47712 
9.84510 

9.30103 

O.32222;; 
O.2O4I2 

9-95424 
0.04139 

0.  6O2O6  M 

9.00000;; 
0.30103 

log  aa 
0.15836 
0.95424 
9.69020 

log  ab 

9.38021;; 

o.  79934'* 
0.04922 

log  al 
0.03342;; 

0.51851 

log  as 
9.07918 
0.77815 

log  bb 
8.60206 
0.64444 
0.40824 

log  l>l 

0.36361;; 
0.806  1  8« 

log  i>s 
0.62325;; 

log// 
9.90848 
0.08278 
1.20412 

log/j 
8.95424;; 
0.34242 
0.83251 

and  the  numbers  corresponding  to  these  logs,  are  exactly 
the  same  as  those  in  form  (a).  The  remainder  of  the  com- 
putation is  the  same  as  there  given. 


154 


THE  ADJUSTMENT  OF   OBSERVATIONS. 


Form  (c).   If  we  wish  to  use  a  table  of  squares  altogether, 
then  since 

ab      --%\(a-\-l>y-a-  —  lf\ 

and  therefore 

[«fl=itf(*  +  *)T-  [a*]  -[**]}  (i) 

we  form  the  square  sums 


/)•], 


[//],[(/+  *)'] 

and  perform  the  necessary  subtractions. 

In  doing  this,  first  take  from  the  table  of  squares  the 
squares  aa,  bb,  ...//,  ss,  and  sum  them  ;  next  write  the  co- 
efficients a  of  A-  on  a  slip  of  paper  and  carry  them  over  the 
coefficients  of  y,  3,  .  .  .,  forming  the  sums  <?,-)-  /;,,«,-{-  £,,  •  •  •; 
a,  -\-b»  tfo-f-^.,,  •  •  •  Take  out  the  squares  of  these  numbers 
and  sum  them.  Proceed  similarly  with  the  coefficients  of 
y,  z,  .  .  .  Finish  as  indicated  in  (i). 

Thus  in  the  preceding  example, 


aa 

M 

// 

JJ 

1.44 

0.04 

o.Si 

O.OI 

9.00 
0.49 

10.93 

4.41 
2.56 

I  .21 

16.00 

4.00 

2.89 

7.01 

18.02 

6  .  90 

a  +  b 

(«H-tf 

a  +  l 

(a  +  I? 

a  +  s 

(a  +  s)" 

I.O 

i  .00 

0.3 

0.09 

1-3 

1  .69 

0.9 

0.81 

4.1 

16.81 

5-0 

25.00 

•2-3 

5-29 

3-3 

10.89 

I.O 

I.OO 

7.10 

27.79 

27.69 

[aa]  + 

[W]  =  I7.94 

[aa\  +  [ 

[aa]  +  [ss] 

=  17.83 

—  10.84 

—  1.16 

9.86 

-    5-42 

—  0.58 

4.93 

=  [*/] 

=    [<"J 

giving  the  same  results  as  before. 


INDIRECT   OBSERVATIONS.  I  55 

This  form,  which  is  very  neat  analytically,  was  first  given 
by  Bessel  in  the  Astron.  NacJir.,  No.  399. 

A  consideration  of  the  simple  case  of  three  observation 
equations,  each  involving  two  unknowns,  will  show  that  to 
form  the  normal  equations,  using  a  log.  table  only,  24  entries 
in  the  table  are  required,  while  by  this  method  we  only 
need  to  enter  a  table  of  squares  18  times,  thus  effecting  a 
saving  of  6  entries.  The  Bessel  method  has  also  the  ad- 
vantage that,  as  we  deal  with  squares,  all  thought  with 
regard  to  sign  is  done  away  with.  Besides,  if  the  table  of 
squares  is  a  very  extended  one,  accuracy  can  be  had  to  a 
greater  number  of  decimal  places  than  with  an  ordinary 
log.  table.  As  compared  with  the  logarithmic  form,  then, 
this  method  is  to  be  preferred,  more  especially  when  the 
coefficients  are  not  very  different. 

On  the  other  hand,  if  Crelle's  tables  or  a  computing 
machine  is  to  be  had,  the  direct  process  explained  in  (a)  is 
much  to  be  preferred  to  either,  as  experience  will  show. 

It  is  worth  noticing  that  whichever  method  of  formation 
of  the  normal  equations  is  adopted,  labor  will  be  saved  by 
changing  the  units  in  which  the  unknowns  are  expressed  if 
the  coefficients  of  the  different  unknowns  are  very  different. 
Thus,  suppose  we  had  the  observation  equations, 

Check  sums. 

looo.r  -(-0.0001^  =  4. 1 1         1004.1101 
999.v  -(-  O.OOO2J/  =  3.93         1002.9302 

from  which  to  find  x  and  y. 
By  placing 

x'  --  i oo.i',  y'  =  o. o \y 

the  equations  reduce  to 

Check  sums. 

i  o.i''     -f-o.oiy  =  4. 1 1      14.12 

9.99.1-'  -f  0.02/  =  3.93  13.94 

which  are  in  more  manageable  shape  for  solution. 


156  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Solution  of  tJie  Normal  Equations. 

Before  beginning  the  solution  of  a  series  of  normal  equa- 
tions we  should  consider  whether  the  object  is  to  find 

(1)  the  unknowns  only,  or 

(2)  the  unknowns  and  their  weights  ; 
and,  in  the  latter  case, 

(a)  whether  the  number  of  unknowns  is  large, 

(b)  whether  many  of  the  coefficients  of  the  unknowns  in 

the  normal  equations  are  wanting. 

Normal  equations  may  be  solved  by  the  ordinary  algebraic 
methods  for  the  elimination  of  linear  equations  or  by  the 
method  of  determinants.  When,  however,  they  are  numer- 
ous the  methods  of  substitution  and  of  indirect  elimination, 
both  introduced  by  Gauss,  are  more  suitable.  Each  has  its 
advantages,  which  will  be  pointed  out  as  we  proceed.  The 
method  of  substitution  is  quite  mechanical,  and  is  well 
suited  for  use  with  an  arithmometer,  which  is  as  great  a 
help  in  solving  as  it  is  in  forming  the  normal  equations. 

86.  The  Method  of  Substitution.—  For  convenience 
in  writing,  take  three  unknowns,  x,  y,  z,  the  process  being 
the  same  whatever  the  number. 

The  normal  equations  are 

]j  +[«>=[«/] 


From  the  first  equation 

[ah]          [tfr]      .    [at] 

_r  —  —  i  —  l<o  —  t  —  -s-\--  —  -  (2) 

[aa\y       \aa\     r  \aa\ 

Substitute  this  value  in  the  remaining  equations,  and,  in 
the  convenient  notation  of  Gauss,  there  result 

\bb.i\y  +  \bc.\\s  =  [/;/.  i] 

>=|W.i]  (3) 


INDIRECT   OBSERVATIONS.  157 

where 

\bb.i  ]  =  m  -[^[^.1 

\aa\ 


(4) 


Again,  from  the  first  of  equations  3, 

_[^i]          [*/.i] 
[W.i]~   r[^.i] 

which  value  substituted  in  the  second  equation  gives 

»=t^l  (6) 


where 


(7) 


Having  thus  found  ^,  we  have  y  at  once  by  substituting  in 
(5),  and  thence  .r  by  substituting  for  y  and  z  their  values 
in  (2). 

The  first  equations  of  the  successive  groups  in  the  elimi- 
nation collected  are 


\CC.2\Z  =\cl.2\ 

These  are  called  the  derived  normal  equations. 


158  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Divide  each  of  these  equations  by  the  coefficient  of  its 
first  unknown,  and 


i    \a"  \       ,    \ac\ 

x  H-  -     -  y  -f       -  .?  — 


[<7rt]          [««]          [««] 

[^<:.  ij       _  [^/.  i]  /  N 

p.l]"  =[^7i] 


;  87.  Controls  of  the  Solution. — In  solving  a  set  of 
normal  equations  a  control  is  essential.  It  is  sometimes 
recommended  to  solve  the  equations  arranged  in  the  reverse 
order,  when,  if  the  work  is  correct,  the  same  results  will  be 
found  as  before.  But  what  is  wanted  in  a  control  is  a 
means  of  checking  the  work  at  each  step,  and  not  at  the 
conclusion,  it  may  be,  of  several  weeks'  work,  when,  if 
the  results  do  not  agree,  all  that  is  known  is  that  there 
is  a  mistake  somewhere  without  being  able  to  locate 
it. 

(a)  Continuation   of  the  formation  control.     Experience 
has  shown   that  it  is  convenient  to  carry  on  through  the 
solution  the  check  used  in  forming  the  equations.     It  merely 
consists  in  placing  as  an  extra  term  to  each  equation  the 
sums  \as],  [bs],  .  .  .  \_ls~],  and  operating  on  them  in  the  same 
way  as  on  the  absolute  terms  [#/],  [#/],  .  .  .    The  sum  of  the 
terms  in   every  line,  after  each  elimination  of  an  unknown, 
must  be   each   equal  to   the    check   sum   numerically  ;    the 
closeness  of  the   agreement  depending  on  the    number  of 
decimal  places  employed. 

This  check  may  be  applied   at  every  step  and  mistakes 
be  weeded  out. 

(b)  The  diagonal  coefficients  \aa\,  \_bb~],  ...  of  the  normal 
equations,  and  [aa~\,  \bb.  i],  \_cc.2~],  .   .  .  of  the  derived  normal 
equations,  are  always  positive. 


INDIRECT   OBSERVATIONS.  159 

For  [aa],  \bb.  i],  .  .  .  being  the  sums  of  squares,  are  posi- 
tive. Also 

Mc»-'Hsj:E»]  =  £':'+££ 

a  positive  quantity. 

Similarly  for  \cc.2\  [dd.$\t  .  .  . 

The  principle  may  be  of  use  as  a  check  in  the  solution 
of  a  series  of  normal  equations  which  are  apparently  cor- 
rect, but  which  'have  been  improperly  formed.  The  follow- 
ing normal  equations,  which  came  up  once  in  my  own  ex- 
perience, will  show  this  : 

7,r+    77-9.1,2=26 

jx  +  287  —  1 2.3-  =64 

-  9-i.r  —  127+ 12^  =    -39 

The  derived  normal  equations  are 

jx  +    77  —  9.  i  s  =  26 
+  217  —  2.9^=38 


showing  by  the  presence  of  the  negative  diagonal  term  that 
there  was  a  mistake  somewhere.  An  examination  of  the 
data  from  which  the  normal  equations  were  derived  showed 
a  condition  wanting  and  a  condition  incorrectly  expressed. 
After  the  proper  corrections  had  been  made  the  solution 
was  carried  through  without  trouble. 

(c)  By  equations  8,  Art.  Si,  the  residuals  found  by  sub- 
stituting for  x,  y,  s  their  values  in  the  observation  equations 
must  satisfy  the  relations 

[ar]  =  [bi>~\  =  .   .   .  =  o 

(d)  A  very  complete   check   is  afforded  by  the  different 
methods   of  computing  [IT],  the  sum  of  the  squares  of  the 
residuals.     (See  Art.  100.) 

88.  Forms  of  Solution. — In   applying  the  method  of 
substitution  to  any  special   example  it  is  important  that  the 


:6o 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


arrangement  of  the  computation  be  convenient  and  that 
every  step  be  written  down.  Experience  teaches  that  sim- 
plicity and  uniformity  of  operation  are  great  safeguards 
against  mistakes. 

Form  (a).   Solution  without  logarithms. 

The  following  form  has  been  found  by  experience  to  be 
convenient.  It  is  well  fitted  for  use  with  the  arithmometer 
or  any  other  rapid  method  of  multiplication.  The  form 
can  be  readily  modified  to  suit  computer's  tastes. 

For  illustration  let  us  take,  as  before,  three  unknowns, 
x,  y,  z.  The  computation  is  divided  into  sections,  each  sec- 
tion being  formed  in  a  precisely  similar  way,  and  in  each 
section  one  unknown  is  eliminated. 

Given  the  normal  equations, 


No. 

.r 

y 

2 

Check. 

Remarks. 

I. 
II. 
III. 

\aa\ 

H 
[at] 

Ml 

\bb} 
[be] 

ac\ 
bc\ 
cc\ 

[«/] 
ten 

[el] 

as] 
faj 
«] 

Solution. 


IV. 

I 

M 

[«*] 

[«£] 
[M] 

f««l 

i.*M 

V. 

II. 

[66] 

[ac] 

t*flfeH 

T']0 

1  fcl 

IV.  X  [«*] 

ii. 

VI. 

[66.1] 

[Ar.i] 

[«.i] 

[*r.i] 

II.  -V. 

VII. 

ME^3 

[ac]EffJ 

MT£ 

M~ 

IV.  X  [rtf] 

III. 

VIII. 
IX. 

[6c] 

[«•! 

let 

[«] 

III. 

III.  -VII. 

[*.i] 

[«..] 

[rf.I] 

[«.i] 

i 

^ 

[/'/  .  1  ] 
L^V^.  I  J 

L».i] 

X. 

^••^ffirl 

[fc-xlf&Tl 

^•^[£T] 

IX.  X  [fc.i] 

VIII. 

U-.i] 

U'li 

I«.'x] 

XI. 

[«.a] 

[C/.2] 

[«..] 

VIII.  -X. 

XI.n-[V<r.2] 

i 

[«.*] 

iSf! 

INDIRECT   OBSERVATIONS.  l6l 

From  Eq.  IX. 

v_    -rfr.i]  ,  [*/.i] 
[^.i]^~[^.i] 

From  Eq.  IV. 

[**]        JiMltW 
y\aa\         [aa]~r[aa'] 

To  eliminate  the  first  unknown,  x.     In  the  first  line  write 

the  quotients  LfLJ,  LffJ,  .   .   .   that  is,  the  coefficients  of  the 
\aa\   \aa\ 

first  normal  equation  divided  by  [aa],  the  coefficient  of  x  in 
that  equation. 

The  first  line  is  now  multiplied  in  order  by  [a&],  \_ac\, 
forming  the  second  and  fifth  lines. 

In  the  third  and  sixth  lines  write  equations  II.  and  III. 

The  fourth  line  is  the  sum  of  the  second  and  third,  and 
the  seventh  the  sum  of  the  fifth  and  sixth. 

This  concludes  the  elimination  of  ,r,  and  the  results  in  the 
fourth  and  seventh  lines  involve  y  and  z  only. 

Take  now  these  results  and  proceed  in  a  precisely  simi- 
lar way  to  eliminate  y. 

The  value  of  the  last  unknown,  ,7,  next  results. 

Now  proceed  to  find  y  and  .r.  Thus  substitute  for  z  its 
value  in  the  eighth  line,  and  we  have  y  ;  and  for  y  and  z  their 
values  in  the  first  line,  and  we  have  x. 

In  carrying  this  solution  into  practice  there  are  three 
points  that  deserve  special  notice: 

(i)  In  order  to  render  the  work  mechanical,  and  so 
lighten  the  labor,  the  number  of  different  operations  should 
be  made  as  small  as  possible.  Instead,  therefore,  of  divid- 
ing by  [aa],  \bb.\~\,  \cc.2],  it  is  better  to  multiply  by  the 
reciprocals  of  these  quantities,  and,  in  order  to  avoid  sub- 
tractions, to  first  change  the  signs  of  the  reciprocals.  We 
shall  then  have  to  perform  only  two  simple  operations — 
multiplication  and  addition.  By  transferring  the  terms 
[«/],  [/>/],  [V/]  to  the  left-hand  side  of  the  equations  before 
beginning  the  solution,  the  values  of  the  unknowns  will 
come  out  with  their  proper  signs. 


1 62 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


(2)  Equations  VI.  and  VIII.  are  the  normal  equations  with 
x  eliminated.     An  inspection   of  them  shows  that  the  co- 
efficients of  the  unknowns  follow  the  same  law  as  the  co- 
efficients of  the    unknowns    in    the   original    normal  equa- 
tions with   respect  to  symmetry  of  vertical  and  horizontal 
columns.     Hence  in   the    elimination  it  is  unnecessary   to 
compute  these  common  terms  more  than  once.     Thus  [bc.i] 
from  Eq.  VI.  may  be  written  down  as  the  first  term  of  Eq. 
VIII.     This   principle  is   of  great   use   in    shortening   the 
work  when  the  number  of  unknowns  is  large. 

(3)  In  a  numerical  example  it  is  evident  that  since  [aa], 
[bb.i~\,  \cc.2~]  do  not  in  general  divide  exactly  into  the  other 
coefficients  of  their  respective  equations,  and  that  only  ap- 
proximate values  of  the  unknowns  can  at  best  be  obtained, 
it  will  give  a  closer  result  to  divide  by  the  larger  coeffi- 
cients and  multiply  by  the  smaller  than  vice  versa.    Atten- 
tion to  this  by  a  proper  arrangement  of  the  coefficients  before 
beginning  the  solution  results  in  a  considerable  saving  of 
labor,  as  the  successive  coefficients  in  the  course   of  the 
elimination  need  not  be  carried  to  as  many  places  of  deci- 
mals to  insure  the  same  accuracy  that  a  different  arrange- 
ment would  require. 

Ex.  To  make  the  preceding  perfectly  plain  we  shall  solve  in  full  the 
normal  equations  formed  in  Art.  85. 

(i)  Write  the  absolute  term  on  the  right  of  the  sign  of  equality,  and  make 
the  check  sum  equal  to  the  sum  of  the  other  terms  in  each  horizontal  row. 


X 

y 

/ 

Check. 

Remarks. 

I. 
II. 

+  10.93 
-  5.42 

-  5.42 
+  7.01 

+  0.58 
+  853 

+    6.09 

+   10.12 

III. 

+    I. 

-  0.496 

+  0.053 

+     0.557 

I.-H  10.93 

IV. 

II 

V. 

+  1.688 
+  7.01 

-  0.288 
+  8.53 

-     3-019 

+    IO.12O 

III.  x    -  ,4, 
II.  -  IV. 

+  4-322 

+  8.818 

+    13.139 

VI. 

+  I. 

+  2.040  =  y 

+      3-040 

V.  H-  4  322 

VII. 

III. 

VIII. 

+     I. 

-  0.496 
-  0.496 

-   I.  012 
+  0.053 

-      1.508 
+      0.557 

VI.    x     -  c.496 

III.  -  VII. 

+     I. 

o. 

+  1.065  =  x 

+      2.O65 

Hence 


x  =  1. 06 


y  =  2.04 


INDIRECT   OBSERVATIONS. 


(2)  Write  the  constant  term  on  the  left  of  the  sign  of  equality,  and  form  the 
check  so  as  to  make  the  sum  of  the  terms  in  each  horizontal  line  equal  to  zero. 


Reciprocals. 

X 

, 

I 

Check. 

Remarks. 

I. 
II. 

0.0515 

+  10  03 

-   5-42 

-  5.42 
+  7.01 

-  o.s8 

-  8.53 

-  4-93 

+  6.94 

III. 

-    I. 

+  0.496 

*  0.053 

+  0.451 

I.   '    -  0.091°; 

IV. 

II. 

V. 

0.2314 

-  2.688 
+  7.010 

-  0.288 
-  8.^30 

-  2.445 

+  6.940 

III.-    -5  42 
II.  +  IV. 

+  4.322 

-  8.818 

+  4-495 

VI. 

-  i. 

+  2.040  =y 

-  1.040 

V.   «    -  02314 

VII. 

III. 

VIII. 

-  I. 

-  0.456 
+  0.496 

+   I.  012 
+   0.053 

-  o.m6 

->•  o  451 

VI.  '  0.456 

III. 

III.   t  VII. 

-  I. 

+    1.065  =  * 

-  0.065 

In  order  to  find  the  values  of  the  unkno%vns  to  two  places  of  decimals  ihe 
computation  should  be  carried  through  to  three  places,  and  ihe  third  pLce 
dropped  in  the  final  result. 

Form  (b).   The  logarithmic  solution. 

As  an  example  of  the  logarithmic  method  let  us  take  the 
general  form  of  the  preceding  example,  when  R  and  6"  are 
substituted  for  the  absolute  terms  0.58  and  8.53  respec- 
tively. 

In  the  numerical  work  it  is  better  to  convert  all  the 
divisions  into  multiplications.  Therefore  write  down  the 
complementary  logs,  of  the  divisors  with  the  signs  changed. 
Each  multiplier  may  now  be  written  as  needed  on  a  slip  of 
paper  and  carried  over  each  logarithm  to  be  operated  on. 
Thus  for  the  first  operation  the  slip  would  have  on  it 
8.961 38/7,  where  the  ;/  indicates  that  the  number  is  negative. 

Paper  ruled  into  small  squares,  so  as  to  bring  the  figures 
in  the  same  vertical  columns  and  facilitate  additions  and 
subtractions,  renders  the  work  more  mechanical  and  is  con- 
sequently an  assistance  to  the  computer. 

In  general  solutions,  when  the  number  of  unknowns  is 
large,  it  will  be  found  much  better  to  carry  a  double  check, 
one  for  the  coefficients  of  .r,  y,  .  .  .  and  the  other  for  the 
coefficients  of  A',  S,  .  .  .  Though  unnecessary  in  our  ex- 
ample, it  is  inserted  for  illustration. 


164 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


It  will  be  noticed  that  the  coefficients  of  J?,  S  in  the 
values  of  x,  y  follow  the  same  law  of  symmetry  as  the  nor- 
mal equations.  A  little  consideration  will  show  that  this  is 
always  the  case. 

Hence,  attending  to  this,  we  may  shorten  the  computa- 
tion by  leaving  out  the  common  terms.  We  have,  there- 
fore, one  term  less  to  compute  for  each  unknown,  pro- 
ceeding from  the  last  to  the  first.  The  case  is  precisely 
analogous  to  that  of  Art.  83. 


X 

y 

Check. 

R 

S 

Check. 

Remarks. 

I. 
II. 

1093 

-     5.42 

-  5  42 
+  7.01 

-  5-5' 
-  1-59 

-  l. 

-  i. 

+  i. 
+  i. 

III. 

IV. 

i.  03862 
(8.96i38«) 

0.73400« 
9.69538 

o.74H5« 
9.70253 

0. 

8.96138 
+  0.091 

o. 
8.96is8« 
-  0.091 

log  I. 
III.  -  log  10.  93 

Nos. 

V. 
VI. 

II. 

VII. 

o.42938« 
-  2.6t3 
+  7.010 

o.436s3« 
-  2.732 
-  1-590 

9.69538« 
-  0.496 

-  i. 

9.69538 
+  0.496 
+  i. 

IV.  -  log  5.42 
Nos. 
II. 

VI.  +  II. 

+  4-322 

-  4-322 

-  0.496 

-  i. 

+  1496 

vn;. 

IX. 
X. 

063568 
(9-36432») 
-  i. 

o.6}568» 
o.ooooo 
+  i. 

0.69  548« 
9.05980 
+  0.115 

0.0. 

9-36432 
+  o  231 

0.17493 
9-  53925« 
-  0.346 

log  VII. 
VIII.  -  log  4.322 
Nos. 

XI. 
XII. 

8.75518 
+  0.057 
+  0.091 

9.05970 

+  0.115 

9.23463« 
-  0.172 
-  0.091 

IX.  +  log  5'42 
Nos.       JO-93 

XIII. 

-  1. 

+  i. 

+  0.148 

+  0.115 

-  0.263 

.'.  x  =  o.i4§R  +  0.1155 
y  =  0.115/1'-)-  0.2315 

Substituting  for  R  and  S  their  values,  we  have,  as  before, 

x  =  i.  06 
y-  2.04 

Ex.  i.   In  the  elimination  of  n  normal  equations  by  the  method  of  substitu- 
tion, show  that  the  total   number  of  independent  coefficients  in  the  original 

5) 


. 
and  derived  normal  equations  is  —     —  - 

[The  sum  is  \  \  i  .  4  +  2  .  5  +  .  .  .  n(n  +  3)|] 

Ex.  2.   If  the  elimination  of  the  unknowns  in  the  normal  equations  is  car- 
ried out  by  the  method  of  substitution,  the  product 

[aa]  [M.i\  [fc.2\  .   .  . 
hns  the  same  value  whatever  order  has  been  followed. 


INDIRECT   OBSERVATIONS.  165 

[For  it  is  the  determinant 

[at], 


89.  The  Method  of  Indirect  Elimination.  —  This 
method  is  often  of  service  when  the  number  of  unknowns  is 
large  and  many  of  the  terms  in  the  normal  equations  are 
wanting.  The  principle  involved  in  the  solution  is  very 
simple.  If  x',y',  3'  are  approximate  values  of  x,  y,  z,  and 
x\*  y\i  si  tne  corrections  to  these  values,  so  that 

x  =  x'  -f-  x, 

y=y'  +7, 
„  _  ,_/  i  ^ 
-"  —  '  i  *i 

then,  substituting  these  values  in  the  normal  equations,  we 
have 

C*^,  =  [*/], 


where  the  new  absolute  terms  [#/]„  [£/]„  [//],  will  be,  on  the 
whole,  smaller  than  the  original  terms  [a/],  [£/],  [r/].  A 
second  approximation  will  tend  to  decrease  the  absolute 
terms  still  farther.  The  approximations  are  continued  till 
the  absolute  terms  either  vanish  or  are  sufficiently  small. 
Then 

x  —  x'-^x"-^  .  .  .;  y—y'-\-y"-\-  ...;... 


In  finding  the  approximate  values  it  is  best  to  consider 
one  unknown  at  a  time,  and  preferably  the  equations  should 
be  operated  upon  in  the  order  of  magnitude  of  the  absolute 
terms.  Thus  suppose  [a/]  the  largest  absolute  term  ;  then, 
since  the  diagonal  term  is  in  general  the  important  term  in 
a  normal  equation,  we  may  neglect  all  but  it  in  the  equation 


[aa] 
Substitute  for  x  this  value  in  the  three  equations,  when  a 


and  call  the   first   approximation    to   the   value   of 

[aa] 


l66  THE   ADJUSTMENT   OF   OBSERVATIONS. 

value  of  y  or  of  z  may  be  similarly  found.  Or  the  form 
of  the  equations  may  be  such  that  first  approximations  to 
the  values  of  both  x  and  y,  or  of  x,  y,  and  s,  may  be  ad- 
vantageously found  from  the  original  equations  before 
making  any  substitutions.  These  and  similar  points  must 
be  decided  according  to  the  circumstances  of  the  case  in 
hand. 

If  the  diagonal  terms  in  the  normal  equations,  instead 
of  being  much  larger  than  the  other  terms,  are  but  little 
larger  than  several  of  them,  considerable  difficulty  will 
be  found  in  making  the  approximations.  An  approxima- 
tion made  for  one  unknown  may  counterbalance  those 
made  for  several  others,  and  the  whole  process  will  be 
found  tedious  and  troublesome.  Various  expedients  have 
been  suggested  for  getting  over  this  difficulty  ;  but  in  all 
cases  where  the  normal  equations  are  not  very  loosely 
connected  (that  is,  where  many  terms  are  wanting)  and 
the  diagonal  coefficients  large,  my  experience  has  been 
that  it  is  much  better  to  use  the  method  of  substitution, 
or,  in  simple  cases,  the  ordinary  algebraic  methods  of 
elimination. 

While  employed  on  the  adjustment  of  the  primary 
triangulation  of  Lake  Superior  we  came  across  Von  Free- 
den's  glowing  account*  of  the  advantages  of  this  method 
of  solution,  and  determined  to  give  it  a  trial.  The 
number  of  equations  to  be  solved  was  32.  Where  the 
connection  of  the  unknowns  was  loose  it  worked  well 
enough,  but  where  they  fell  in  groups  it  was  very 
troublesome  and  slow.  On  the  whole,  the  work  was 
more  fatiguing  and  took  longer  time.  We  never  tried 
the  method  again,  though,  as  it  works  so  nicely  in 
simple  cases,  we  were  at  one  time  very  much  in  its 
favor. 

Gauss,  the  author  of  this  form  of  solution,  describes  it  as 
in  general  tedious  (langwierig),  and  very  justly.  For  the 
contrary  view  see  Coast  Survey  Report,  1855,  App.^f;  Von 

*  Die  Praxis  dcr  Mcthodc  dcr  k2cinsten  Quadrate.     Braunschweig,  1863. 


INDIRECT    OBSERVATIONS. 


I67 


Freeden,  loc.  cit. ;  Vogler,  Grundziige  der  Ausgleicliungsrech- 
nutig,  p.  129. 

Ex.  Required  to  solve  the  equations  formed  in  Art.  85. 

Solution. 


Check. 

_r 

y 

10.93 

-  5-42 

-  5-51 

-  5.42 

+  7-oi 

-  1-59 

—  0.58 

-3.53 

+  9-II 

-  5.420 

+  5-465 

+  7-oi 
-  2.71 

-  1-59 

-  2.755  . 

0-5 

I  .0 

i  .0 

-  0.535 
+  5.465 

-4.230 

-  2.  "I 

+  4-765 

-  2.755 

-  5.420 

4-  7-01 

-  1-59 

0.5 
0.05 

0.03 

-0.490 
+  0.5465 

—  0.1626 

4-  0.070 
—  0.271 
+  O.2IO3 

4-  0.420 

-  0-2755 
-  o  0477 

—  o.  1061 

+  0.0093 

+  0.0968 

->r  0.1093 

—  0.0542 

-0.0551 

O.OI 

—  0.0542 

+  0.0701 

—  0.0159 

0.005 

O.OI 

—  0.0510 
+  0.0546 

4-  0.0252 
—  0.0271 

+  0.0258 
—  0.0275 

+  0.0036 

—  0.0019 

—  0.0017 

1  .065 

2.04 

The  first  two  lines  contain  the  coefficients  of  x,  y,  together  with  the  check 
sums  ;  the  third  line  the  absolute  terms  and  their  check  sum. 

The  first  approximations  are  taken  y=\  and  x  —  o. 5.  The  second  and 
first  lines  are  multiplied  by  these  numbers  and  the  products  written  in  the 
fourth  and  fifth  lines.  The  third,  fourth,  and  fifth  lines  are  summed,  and 
y  =  i,  j;  =  o.5  are  taken  as  the  next  approximations.  The  sum  of  the  x 
column  of  approximations  gives  the  value  of  .r,  and  of  the  y  column  the 
value  of^. 

90.  Combination  of  the  Direct  and  Indirect 
Methods  of  Solution. — A  numerical  example  illustrating 
an  ingenious  combination  of  the  direct  and  indirect  methods 
of  elimination  is  given  in  the  Coast  Survey  Report  tor  1878, 


1  68  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Appendix  8.     For  performing  the  multiplications  necessary 
Crelle's  tables  are  used  altogether. 

In  order  to  make  the  process  employed  readily  followed 
I  will  give  in  general  terms  the  solution  of  the  three  normal 
equations 

\_ad\x  -f  \ab~\y  +  \ac~\s  —  [>/] 

\_ab~\x  +  \bb~\y  +  \bc\z  = 


according  to  this  form. 

The  coefficients  and  absolute  term  of  the  first  equation 
are  written  in  line  i,  Table  A.  The  reciprocal  of  the  diago- 
nal coefficient  [aa]  is  taken  from  a  table  of  reciprocals  and 
entered  in  the  front  column  with  the  minus  sign  prefixed. 
The  remaining  terms  of  line  I  are  multiplied  by  this 
reciprocal,  and  the  products  written  in  line  2.  This  gives 
x  as  an  explicit  function  off  and  z. 

The  coefficients  and  absolute  term  of  Eq.  2  (omitting  the 
coefficient  of  x)  are  written  in  line  i,  Table  B.  The  terms 
in  line  2,  Table  A,  beginning  with  that  under  y,  are  multiplied 
by  \ab\  the  coefficient  of  y,  and  the  products  set  down  in 
line  2,  Table  B.  The  sum  of  lines  i,  2,  Table  B,  is  now  writ- 
ten in  line  3,  Table  A. 

Line  4,  Table  A,  is  found  from  line  3  in  the  same  way  as 
line  2  was  found  from  line  i.  This  gives  y  as  an  explicit 
function  of  z. 

The  coefficients  and  absolute  term  of  Eq.  3  (omitting  the 
coefficients  of  x  and  y]  are  written  in  line  3,  Table  B.  The 
terms  in  lines  2,  4,  Table  A,  beginning  with  those  under  z, 
are  multiplied  by  [ac],  [&M],  the  coefficients  of  z  in  lines  i,  3 
respectively,  and  the  products  set  down  in  lines  4,  5,  Table 
B.  The  sum  of  lines  3,  4,  5,  Table  B,  is  written  in  line  5, 
Table  A. 

Line  6,  Table  A,  gives  the  value  of  z. 

The  next  step  is  to  find  y  and  x.  The  coefficients  of 
the  explicit  functions  are  written  in  Table  C.  The  abso- 
lute terms  of  the  explicit  functions  are  written  in  the  first 


INDIRECT    OBSERVATIONS.  169 

line  of  Table  D.  The  value  of  z  is  multiplied  by  the 
coefficients  of  z  in  Table  C,  and  the  products  written  in 
the  second  line  of  Table  D.  The  sum  of  the  numbers  in 
column  y  gives  the  value  of/  written  underneath  in  line 
3.  The  value  of  y  is  multiplied  by  the  coefficients  of  y  in 
Table  C,  andx  the  products  written  in  the  third  line  of 
Table  D.  The  sum  of  the  numbers  in  column  x  gives  the 
value  of  x. 

The  values  of  x,  y,  z  are  now  found  to  three  places 
of  decimals.  Denote  them  by  x',  y',  z'.  If  these  values 
are  not  sufficiently  close  a  second  approximation  must  be 
made.  This  we  proceed  to  describe. 

First  substitute  the  values  obtained  in  the  original  nor- 
mal equations,  and  carry  out  to  a  sufficient  number  of  deci- 
mal places.  The  residuals  are  written  in  the  first  line  of 
Table  E.  The  coefficients  in  line  i,  Table  C,  are  multiplied 
by  —  [#/],,  and  the  products  written  in  line  2,  Table  E. 
The  first  reciprocal  in  Table  A  is  multiplied  by  the  same 
residual,  and  the  product  written  in  column  x,  line  I, 
Table  F.  The  sum  of  the  numbers  in  column  2,  Table  E,  is 
written  underneath,  as  —  \bl.  ij,. 

The  coefficient  in  line  2,  Table  C,  is  multiplied  by 
-  \bl.  i],  and  the  product  written  in  line  3,  Table  E.  The 
second  reciprocal  in  Table  A  is  multiplied  by  the  same 
residual,  and  the  product  written  in  column  y,  line  i,  Table  F. 
The  sum  of  the  numbers  in  column  3,  Table  E,  is  written 
underneath,  as  —  [V/.2],. 

The  third  reciprocal  of  Table  A  is  multiplied  by  this 
residual,  and  the  product  written  in  column  z,  line  i, 
Table  F.  This  gives  the  correction  to  the  value  of  z. 
The  first  line  of  Table  F  corresponds  to  the  first  line 
of  Table  D,  and  exactly  the  same  process  employed  in 
Table  D  will  complete  Table  F.  The  total  values  now  are 


y=y 

.  —  «/  i 

"*     ••'    i 


I/O  THE   ADJUSTMENT   OF   OBSERVATIONS. 


B 


Recip. 


[W.i] 


X 

y 

z 

+  [-] 

+  [«*] 

+  [-] 

-M 

*  = 

[ab] 
[aa] 

[at] 

^&! 

[W.i] 

+  [fc-.l] 

-[W.i] 

y- 

"  [*£i] 

+  [«£] 

z 

[rf.3] 
1        [<V.2] 

[a*] 


[be] 


D 


.V 

z 

« 

y 

z 

-0 

[at] 
[aa] 

+  [<l 

+  [*dil 

+  [£^1 

[fci] 

z' 

"[Ail] 

"  [aa]£/ 

"  [S]2' 

-  ra^]J/' 

y 

*' 

E 


- 

, 

z 

X 

y 

z 

-M, 

-[W], 

-M, 

+   [«/], 

[W.i], 

[rf2]] 

[a*] 

[ac] 

[aa] 

'  [W.i] 

+    [«-2] 

+  L^][a'J] 

+  T^l[a/]l 

~    r  V" 

£" 

[W.i], 

[if  l] 

[aa] 

[W.i]- 

+  [All][W'l]' 

_  f?y  „ 

y" 

-    [c'  2] 

[aa] 

jr" 

INDIRECT   OBSERVATIONS.  \J\ 

Addition  of  New  Equations.  —  It  often  happens  that  after 
the  adjustment  of  a  long  series  of  observations  additional 
observations  are  made  leading  to  additional  condition  equa- 
tions. To  make  a  solution  de  noi'o  is  necessary,  but  the 
work  may  be  very  materially  shortened  by  the  process  just 
given.  Suppose,  for  simplicity,  that  one  new  condition  has 
been  established.  This  will  give  one  additional  normal 
equation,  which  may  be  written 

\_ad~\x  +  \bd~\y  +  \cd~\z  +  \dd]w  =  \dl]  (i) 

w  being  the  new  unknown. 

The  extra  term  to  each  of  the  other  normal  equations 
may  be  written  down  at  sight.  The  complete  equations  are 

\aei\x  +  \ab\y  +  \ac\z  +  [ad]w  =  [a/] 


-f-  [_dd~\iv  =  [til] 

Now,  values  of  x,  y,  z  have  been  already  found  from  the 
normal  equations  resulting  from  the  original  condition 
equations,  and  these  values  may  be  taken  as  first  approxi- 
mations to  the  values  of  x^y,z  resulting  from  the  above 
four  normal  equations.  Substitute  in  (i),  and 

\_ad~\x  +  \bd~\y  +  \cd~\s'  +  \dd~\w  =  [<*/]'  (2) 

where  .r',  y'  ,  z  are  corrections  to  the  approximate  values  of 
.v,  }',  z.  The  solution  is  now  finished  as  follows: 

Form   Table  C  (a)   by  adding   the  extra    column  w  to 

Table  C.     The  term  -     '^j  is  found  by  multiplying  \_ctd  \  by 

the  first  reciprocal.  The  coefficients  of  the  new  equation, 
(2),  are  written  in  the  first  line  of  Table  G.  Since  correc- 
tions to  values  already  found  are  required,  the  method  of 
proceeding  must  be  similar  to  that  employed  in  Table  E. 
The  notation  in  Tables  C  (a)  and  G  explains  this. 

The  reciprocal  of  the  sum  of  column  :;',  that  is,  \dd.^\,  in 
23 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


Table  G  is  written  last  in  the  column  of  reciprocals  of 
Table  C  (a)  with  the  minus  sign.  The  product  of  this 
reciprocal  and  the  absolute  term  —  \dl'\  of  the  new  equation, 

that  is,  A    -.,  is  an  approximate  value  of  w.     This  value  of  w 

is  multiplied  by  the  terms  in  the  last  column  ot  Table  C  (a), 
and  the  products  are  written  in  the  first  line  of  Table  II. 
Column  s  gives  the  correction  to  z.  Table  II  is  now  com- 
pleted in  the  same  way  as  Tables  D  and  F. 

C  (a)  G 


y 

z 

. 

I 
[oa] 

[a*] 
[aa] 

[ac\ 
[aa] 

M 
[aa] 

i 
~  [**T] 

[fer.i] 
[W.i] 

[M.i] 
[W.i] 

i 

"  i^.2] 

[frf.2] 

'tec.  2] 

1 

"  JVW.3] 

X 

y 

• 

7W 

M 

CM] 

[«g 

[rfrf] 

-  KH 

-  Sarf] 

[arfj 

fiZT] 

[*C'1]  M  i] 

[M''    M 

[W.i] 

[W.ij 

M.2] 

_    [f^-.2],frf  2] 

ftu.ti 

[ad] 


H 


[W.i] 

- 


[Cd2] 
;.•          ,7 

[cc.a] 


91.  Time  required  to  Solve  a  Set  of  Equations.— 

The  labor  involved  in  solving  a  series  of  normal  equations, 
and  the  consequent  time  employed,  increases  enormously 
with  an  increase  in  the  number  of  normal  equations.  To 
any  one  who  has  never  been  engaged  in  such  work  it  will 


INDIRECT   OBSERVATIONS.  173 

seem  out  of  all  reason.  Thus  Dr.  Hug-el,'- of  Hessen,  Ger- 
many, states  that  he  has  solved  10  normal  equations  in  from 
10  to  12  hours,  using  a  log.  table,  but  that  29  equations  took 
him  7  weeks. 

The  following  are  examples  of  rapid  work:  Gen. 
Baeyer,  in  the  Kiistcnvcrmessung  (Vorwort,  p.  vii.)  mentions 
that  Herr  Dase  solved  86  normal  equations  between  the 
first  of  June  and  the  middle  of  September;  and  Mr.  Doo- 
little,f  of  the  U.  S.  Coast  Survey,  solved  41  normal  equa- 
tions in  5^  days,  or  36  working  hours. 

A  great  deal  depends,  so  far  as  speed  is  concerned,  on 
the  form  of  solution  and  on  the  mechanical  aids  used. 
With  a  machine  or  with  Crelle's  tables  much  better  time 
can  be  made  than  by  the  logarithmic  method,  which  is  by 
far  the  most  roundabout.  Mr.  Doolittle  used  Crelle's  ta- 
bles and  the  form  explained  in  the  preceding  article. 

In  comparisons  such  as  these  it  is  important  to  know 
how  many  of  the  terms  of  the  normal  equations  are  want- 
ing. Herr  Dase's  and  Mr.  Doolittle's  equations  were  both 
derived  from  triangulation  work.  The  latter  had  only  430 
terms  in  his  41  equations,  while  the  former  had  3141  differ- 
ent terms  in  his  86  equations. 

A  machine  for  the  solution  of  simultaneous  linear  equa- 
tions has  been  invented  by  Sir  \V.  Thomson,  of  Glasgow, 
Scotland.  The  results  are  obtained,  by  a  series  of  approxi- 
mations, to  any  accuracy  required.  A  description  of  the 
machine  and  of  the  mathematical  principles  underlying  its 
construction  will  be  found  in  Thomson  and  Tait's  Natural 
PJiilosopJiy,  vol.  i.  Appendix  13.  Sir  W.  Thomson  says: 
"  The  exceeding  ease  of  each  application  of  the  machine 
promises  well  for  its  real  usefulness,  whether  for  cases  in 
which  a  single  application  suffices  or  for  others  in  which 
the  requisite  accuracy  is  reached  after  two  or  three  or  more 
of  successive  approximations." 

With    regard   to  a   machine   for  performing   multiplica- 

*  C,i-nfral-Bcricht  iiber  die  curofaisrhen  Cr,i,h»es:siin^.  iS'.j.  p.  109. 
tAY/,>r/  r.  .V.  Co.tsf  Surrey,  1878,  Appendix  S. 


1/4  THE   ADJUSTMENT   OF   OBSERVATIONS. 

tions  and  divisions,  I  am  certain  that  no  one  will  ever  prefer 
to  use  a  log.  table  in  solving  a  set  of  equations  by  the 
method  of  substitution  who  has  ever  used  any  of  the  forms 
of  arithmometer. 

92.  For  Jacobi's  method  of  elimination,  and  the  applica- 
tion of  determinants  to  the  solution  of  normal  equations,  see 
Astron.    Nachr.,  404,    1960;  Month.   Not.  Roy.  Astron.    Soc., 
vols.  xxxiv.,  xl.;  Proc.  Amer.  Assoc.  for  Adv.  of  Science,  1881. 

The  Precision  of  the  Most  Probable  (Adjusted}  Values. 

93.  The  problem  now  before  us  is  to  find  the  m.  s.  e.  of 
the    unknowns  x,  y,  ...   as  determined    from  a  series  of 
normal   equations.     If  the   observation    equations   are   re- 
duced to  the  same  unit  of  weight,  which  we  shall  take  to 
be  unity  for  convenience,  the  general  form   of  the  normal 
equations  is 

\ad\x  -(-  \ati\y  -\-  .   .   .   =[_al'] 

Wx  +  \bb-\y  +  .   .   .   ==[*/]  (i) 


Let  fjt=  the  m.  s.  e.  of  a  single  observation. 
/jtx,  (jLy,  .  .  .  =  the  m.  s.  e.  of  x,  y,  .  .  . 
px,  /,,...  =  the  weights  of  x,  y,  .  .  . 

From  Art.  56  we  have 

P*ti=P3ti  =  ...=//  (2) 

In  order,  therefore,  to  determine  /Jix,  [Jty,  .  .  .  we  must  make 
two  computations,  one  of  the  weights  px,  py,  .  .  .  and  the 
other  of//,  the  m.  s.  e.  of  a  single  observation. 

It  is  evident  from  an  inspection  of  the  normal  equations 
that  ^r,  y,  .  .  .  are  linear  functions  of  /„  /2,  .  .  .     Let,  then, 
x  —  aj,  -f  «/2  -f  .  .  .  -f  ajn  =  [«/] 

y  =  M  +  M  +  •  •  .  +  /U  =  [/3/]  (3) 


in  which  «„  «2,  .  .  .  ;  &,  /9a,  ...;...  are  functions  of 
alt  ba  .  .  .;«„,&„.  .  .  ;  .  .  .  their  values  being  as  yet  un- 
determined. 


INDIRECT   OBSERVATIONS.  175 

Now,  n  being  the  m.  s.  e.  of  each  of  the  observed  quanti- 
ties J/,,J/,,  .  .  .  J/M,  must  be  also  the  m.s.  e.  of  /„/„,  .../„, 
which  differ  from  Mlt  Mv  .  .  .  Mn  by  known  amounts  (see 
Art.  81).  Hence  since  /„  /.,,...  /„  are  independent  of 
each  other  (Art.  19), 

H*—  /*"[««],  ti  =  l?[tf~\,  •   •   •  (4) 

and  therefore 


We  shall  first  of  all  determine  the  weights/,,/,,  .  .   . 

Before  proceeding  with  the  general  proof  let  us  con- 
sider the  simple  case  of  two  unknowns,  x,y,  which  are  to 
be  found  from  the  three  observation  equations 

a,x  +  bly  =  ll 
anx  -f-  b.j  =  /., 
<*»*  +  b*y  —  I* 
all  of  the  same  weight. 

The  normal  equations  are 

\ad\x  -f-  [afr]  y  =  [«/] 
^ab\-c  -\-\bti\y  =  \bl\ 

Their  solution  by  the  method  of  substitution  gives 


Hence  since  /,,  /„  /,  are  independent  of  one  another, 

=  rrl--,  (6) 


1/6  THE   ADJUSTMENT   OF   OBSERVATIONS. 

that  is,  the  weight  of  y  is  the  denominator  of  the  value  of  y 
found  from  the  solution  of  the  normal  equations  by-tJie  Gaussian 
method  of  substitution. 

The  general  demonstration  may  be  carried  out  more 
simply  by  the  application  of  the  principles  of  undetermined 
coefficients.  Thus  substitute  [«/],  [/?/],  •  •  •  for  x,  y,  .  .  . 
in  the  normal  equations  (i),  and 

M  [«/]  H- [«*][#]+.  •  •  =[>/] 

9/]+.  •  .=[*/]  I  7.1 


or,  arranging  according  to  /,,  /„,... 

I  [««]«,  +  [V^R  +.  •  .  -«,}/, 

+  I  [WK  +  [^R  + 


+  {  [«*]«,  +  WA  +  •  -  •  -  W  +  •  •  • 


Tlie  unknown  quantities  «i;  «.,,  .  .  .  may  be  so  determined 
that  the  coefficients  of  /,,  /g,  .  .  .  shall  each  equal  zero. 
Hence  the  several  sets  of  equations 


< 
( 


i  +  [«^]    +  •  •  .  —  ^  —  o 

,  -f  0/41  +  .   .   .  -  a,  —  o 

...  (8) 


are  simultaneously  satisfied  by  the  same  values  of 


IXfJlRKCT   OBSERVATIONS.  1/7 

Multiply  the  equations  of  each  set  by  a^  a.,,  ...  in  order, 
and  add  ;  then  necessarily 

\aa\  =  i ,  [a ft \  =  o,[ ay  |  =  O,  .   .   .  (y) 

In  a  similar  way,  multiplying  by  />,,  A,,  .  .  .  ;  r,,  r..,  .  .  .,  etc., 
and  adding,  there  result 

[l,a\  =0,  [A?  |=i,  |?/}  =  o,  .  .  . 
\ca \  —  o,  [eft \  =  o,  [cy  ]  =  i ,  .  .  . 


Again,  multiply  the  first  set  by  '/.,,  «.,,  .  .  .,  the  second  by 
flu  fl»  •  •  •>  and  so  on,  and  add,  and  we  have  the  sets  of 
equations 

i     an    aa   --  ab    «9  --  •   •   •  =    '«   =  T 


V. 


(10) 


from  which  equations  [««],  [«^|,  .   .    .    may  be  found. 

It  is  plain  that  the  coefficients  of  [««],  [a^J,  .  .  . 
1/^5],  ...  in  these  equations  are  the  same  as  those  ot 
-i',  y,  ...  in  the  normal  equations,  and  that  the  absolute 
terms  are  i,  o,  ...;  o,  i,  ...;...  instead  of  [a!  ], 
|/>/],  .  .  .  Hence, 

94.  To  Find  the  Weights  of  the  Unknowns.—/;/ 
tlic  first  normal  equation  write  \  for  a  I  j,  and  in  the  otlier 
normal  equations  put  o  for  each  of  />/],[<•/,  .  .  .  ;  the  value 
of  x  found  from  these  equations  will  be  the  reciprocal  of  t lie 
weight  of  x,  and  t  lie  values  of  y,  z,  .  .  .  will  be  the  values  of 
[«^],  ['/;•'],  .  .  .  In  the  second  normal  equation  ft/rite  i  for\bl  , 
and  in  the  other  equations  put  o  for  each  of  al  ,  \cl  \ •  tin- 
value  of  y  found  from  these  equations  will  be  the  reciprocal  of 
the  weight  of  y,  and  the  values  of  z,  .  .  .  found  will  be  the 


1 78 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


values  of  [$K],  .  .  .  Similarly  for  each  of  the  unknowns  in 
succession.  For  example,  the  weight  equations  for  three 
unknowns  are 


[««] 

W] 

[«y] 

<-  [aa] 

+  [a*] 

+   [ac] 

i 

>-  [a*] 

+  [A*] 

f     [*C] 

o 

i   [ac] 

+  M 

+  M 

o 

[«fl 

[J3/3J 

[W 

+  [aa] 

+  [a*] 

+  [ac] 

o 

+  [a*] 

+  [W] 

+•  [Ac] 

I 

+  [af] 

+  [Ac] 

+  [cc] 

0 

[•»] 

[6y] 

[VYl 

+  [aa] 

+  [a*] 

t  [ac] 

o 

+  [a*] 

+  [W] 

+  [Ac] 

•    0 

+  M 

+  M 

+  [cc] 

I 

The  quantities  [«;9],  [«^],  •  •  .  are  necessary  when  the 
weight  of  a  linear  function  of  the  unknowns  is  required,  as 
will  be  seen  presently.  (See  Art.  101.) 

It  is  evident  from  the  form  of  the  weight  equations  that 
if  the  elimination  is  carried  through  by  the  method  of  sub- 
stitution, the  successive  steps  to  the  left  of  the  sign  of 
equality  are  the  same  as  in  Art.  88.  Hence  if  the  equations 
are  arranged  so  that  the  unknown  whose  weight  is  re- 
quired— xr,  for  example — is  found  first,  we  should  have  the 
forms 


X 

y 

z 

[aa] 

+  [ai] 

\-  [ac] 

[a/] 

+  [*ft.i] 

f  [if.  i] 

[W.i] 

+    [CC.2] 

[Cl.2] 

.'.  [«.2]s  =  [r/.2] 

[«rl 

[Oy] 

[yy] 

[aa] 

+  [a*] 

+  [ac] 

0 

H-  [AA.i] 

+  [Ac.:] 

0 

+  Ice.  2^ 

I 

•  '•  ["••2][yyj  •-  i 

Hence  the  coefficient  of  the  unknown  first  found  in  the 
ordinary  solution  of  the  normal  equations  is  the  weight  of 
that  unknown.  By  a  separate  elimination  for  each  unknown 
the  weight  of  that  unknown  could  be  found  as  above,  but 
the  process  would  be  intolerably  tedious. 

95.  Special  Cases  of  Two  and  Three  Unknowns. — We  may, 
however,  from  the  preceding  derive  formulas  for  the  weights 


INDIRECT   OBSERVATIONS.  179 

in  a  series  of  normal  equations  containing  not  more  than 
three  unknowns,  which  are  easy  oF  application. 

Thus  with  two  unknowns,  x  and  y,  y  being  found  first, 


[aa] 
In  the  reverse  order,  x  being  found  first, 


or 


where 


With    three   unknowns,  x,  y,  rj,  performing  the   elimina- 
tion of  the  normal  equations  in  the  order  ,7,  y,  x,  we  have 


\cc.i\ 


-\bU\cc\-\bc\bc\ 
which  expressions  are  easily  transformed  into 

A  =  ,      \\hb\-\t  il>\' 

iy        i  .,„  ii  ....  i        HTTp* 


where 

/  =  [^]1  M>\\cc\  -f  2\<il>}\lx--}\n(-\  -  \aa\bc\-  -  \l>l>\\<ic\>  -  \cc\ 


180  THE   ADJUSTMENT   OF   OBSERVATIONS. 

From  these  formulas  the  weights  of  the  unknowns  can  be 
found  directly  without  solving  the  normal  equations.  If 
the  normal  equations  have  simple  coefficients  it  is  much 
more  rapid  to  find  the  weights  in  this  way  and  solve  the 
equations  by  ordinary  algebra  rather  than  by  the  Gaussian 
method.  But  when  the  number  of  unknowns  exceeds  three 
this  becomes  too  cumbersome. 


Ex.  To  find  the  weights  of  the  adjusted  angles  in  Ex.  4,  Art.  83. 
Here 

A  =  12  X  ii  X  15  —  12  X  16  —  ii  X  49 

=  1249 
and 

1249 

Px  —  -  =  8.4 

149 

1240 

/v  =  —  —  =  Q  «; 
131 


If  ux,  uy,  iiz  denote  the  reciprocals  of  px,  py,  pz  respectively,  then 


Uy    =  O.IO49 

Uz  —  O.IO57 

96.  Modification  of  General  Method.  —  To  carry  out  the 
method  of  Art.  94  directly  as  stated  would  be  excessively 
troublesome,  and  various  modifications  have  been  proposed. 
The  following  scheme,  which  consists  in  running  the  weight 
equations  together,  will  be  found  very  convenient. 

Take,  for  simplicity  in  writing,  three  unknowns,  x,  y,  r, 
and  to  the  ordinary  form  of  the  normal  equations  as  ar- 
ranged for  solution  add  the  columns 

i  o  o 
o  i  o 
o  o  i 

the  check  being  carried  throughout. 


INDIRECT    OBSERVATIONS. 


181 


Perform  the  elimination  exactly  as  stated  in  Art  88,  and 
find  the  values  of  the  unknowns  in  the  usual  way.  We 
have  then 


X 

y 

X 

* 

s 

r 

Check. 

[aa] 

M 

[ac] 

[ail 
\bb\ 
[be] 

n 

[an 

bl 

f        * 
| 
} 

o 

i 
o 

0 
0 

I 

[as]  +  i 

fej  :  ; 

i 

[a*] 

[ac] 

[a/] 

, 

[aa] 

[aa] 

[aa] 

[   ./i.'    1 

[44.  i] 

[6/  .  i  ] 

JP 

i 

k.i] 

tccij 

K'-i] 

f          [ffl 

0 

1 

•j   "[<»<«] 

1 

i 

[*££] 

[«.l] 

I     *i 

! 

[M.ll 

[6*.i] 

LM.i] 

[W.i] 

[<r<r.2] 

[^.2] 

r    "2 

f      S* 

f      I 

I 

[C/.2] 

) 

;      i 

where 


A', 


O  —  —   FT^       ^T  "T~  *J<) 

\bb.i\ 

Now,   taking    the    first   column    in  the  table    under  the 
heading  R,  and  attending  to  Art.  94,  we  have 


A',          \bc.\ 

^Ti-j-^. 

A5 


+  [J.2] 


—  — U  - --  J U        a 

[rtrt]        [W.l]        \cc.2 


1 82  THE  ADJUSTMENT   OF   OBSERVATIONS. 

Similarly  for  the  column  under  S, 


-  I         3 

~\bb.i}^  \_cc.2] 

and  for  the  column  under  T, 

"•=irr]  =  ^ 

Also  it  is  evident  that 


,  .„     , 

"1"  r 


The  forms  of  the  expressions  for  [««],  [/5|5],  [K7],  •  •  •  show 
that  these  quantities  may  be  conveniently  computed  from 
the  preceding  tabular  elimination  scheme.  Thus  the  sum 
of  the  products  of  each  pair  of  numbers  bracketed  under 
the  heads  R,  S,  T  will  give  ux,  uy,  uz  respectively. 

The  convenience  of  this  form  is  seen  in  such  a  case  as 
the  following,  which  is  of  common  occurrence.  In  a  set  of, 
say,  40  normal  equations  the  weights  of  10  of  the  unknowns 
may  be  required.  These  10  would  be  placed  last  in  the 
solution  of  the  equations,  and  the  extra  columns  R,  S,  .  .  . 
added  after  30  of  the  unknowns  had  been  eliminated, 
thus  giving  the  weights  required,  with  a  trifling  increase 
of  work. 


INDIRECT  OBSERVATIONS. 


Ex.  i.  Given  the  normal  equations 


1  2A-  -    73  =  K 

+  i  y  -    4z  =  S 

-    ix  ~    4y  +  153  =  7' 


to  find  the  weights  of  y  and  s. 


X 

y 

z 

S 

T 

+  12 

-  7 

0 

+  II 
-  4 

-  1 

-  4 
+  15 

i 

o 

-  0.5333 

+  ii 

-4 
+  10  9169 

+  '   1 

+  i 

i 

-  0.3636 

+  0.0909  j 

+  9-4^25 

+  0.3636  ) 

*'   i 

i 

+  0.0384  \ 

+  0.1057  1 

Hence 

HZ  —  i  X  0.1057 

Ky     =      I     X    O.O9O9  -f- 

agrceing  with  the  values  in  Art.  95. 


=  0.1057 

X    O.03S4    =    O.IO49 


Ex.  2.   Show  that 


.i]  =  [a/]A'1  +  [M] 
/.2J  =  [,//]  A'.,  +  [_t,l\S.i+  \cl\ 


Lx.  3.   Show  that  the  multipliers  A'i,  A'...,   .  .   .  satisfy  the  conditions 
[rta]A"i  4-  [fl^]=  o 
\na\  ft?  +  \til>}  S-t  +  [<jc]  -  o 


[,ia]A'3  +  [a6]S3  +  [<:{}  7'3  +  \a<i\—  o 
[a/.]  A',  +  [M]S,  +  [/-el  7',  f  [  M\  =  o 
[ac]  A'3  +  [be  ]  S3  +  [ff]  Ti  +  [at}  =  o 


184  THE   ADJUSTMENT   OF   OBSERVATIONS. 

97.  Second  Method  of  Finding  the  Weights  of 
the  Unknowns.  —  If  we  multiply  the  first  of  the  normal 
equations  i,  Art.  93,  by  [«a],  the  second  by  [«/3],  the  third 
by  [a/],  and  so  on  ;  add  the  products,  and  attend  to  equa- 
tions 10,  Art.  93,  we  obtain 


Similarly 

y  =  \ap\\_al-]  +  WW\  +  [MM  +  . 


Hence  since  [««],  [/9/9],  .  .  .  are  the  reciprocals  of  the 
weights  of  x,  y,  .  .  .,  this  method  of  finding  the  weights 
may  be  stated  as  follows  : 

In  any  given  series  of  observations,  Jiaving  formed  the  normal 
equations,  replace  the  numerical  absolute  terms  by  the  general 
symbols  [#/],[#/],  .  .  .  and  find  by  any  method  of  elimination 
the  vahies  of  x,  y,  .  .  .,  in  terms  of  [a  1],  [£/],  .  .  .  /  then  the 
^veight  of  x  is  the  reciprocal  of  tlie  coefficient  of\al~\  in  the  value 
of  x,  the  weight  of  y  is  the  reciprocal  of  the  coefficient  of  \_bl~\  in 
the  value  of  y,  and  so  on. 

The  coefficients  of  the  remaining  symbols  for  the  abso- 
lute terms  in  the  expressions  for  x,  y,  .  .  .  give  the  values 
of  [a/9],  [«7],  .  .  .  ;  [/fy],  ...;...  and  the  numerical  values 
of  the  unknowns  x,  y,  .  .  .  may  be  found  by  substituting 
for  [#/],  \bl~\,  .  .  .  their  numerical  values. 

•  In  this  method  of  computing  the  values  of  the  unknowns 
and  their  weights  a  machine  can  be  used  with  great  ad- 
vantage. 

The  formulas  of  Art.  96  are  easily  derived  from  the  pre- 
ceding principles.  For  solving  the  normal  equations 

\_ad\x  -f  \_ab~\y  +  \_ac\s  —  [>/] 


INDIRECT   OBSERVATIONS.  185 

by   the   method  of  substitution   we  have    for  the   first   un- 
known 


~ 


i    J  •    I-,/-.  _ 

[<r.2]/H  J[«-.2 

Comparing  this  with  the  general  expression  for  x  in  Eq.  i, 


f"x«y,"l  I  I     A  A      T    1  I 


[W.IJ     '     \JC.2] 


Similarly  for  j  and  £. 

£jr.  To  find  the  weights  of  the  unknowns  in  Ex.  4,  Art.  83. 
Solving  the  normal  equations  in  general  terms, 

-r  =  0.1193  [a/]  +0.0224  [I'll  +0.0616  [<•/] 
y  ~  0.0224  [<*t]  +  o.  1049  [/;/]  +  0.0384  [c  /] 
=  0.0616  [«/]+  0.0384  [/'/]+  0.1057  [<-/] 
Hence 

Z'*=  [««]  =  O.II93 

Uy    -    [ft  ft}    =0.1049 

M*  =  [?'r]  =  0.105  7 

as  found  in  Art.  95. 

98.  In  deducing  the  formulas  for  the  precision  of  the 
adjusted  values  in  a  series  of  normal  equations  we  have,  for 
convenience  in  writing,  taken  the  observation  equations  to 
be  reduced  to  weight  unity,  and  the  normal  equations,  con- 
sequently, to  be  of  the  form 

\aa\v  +\ab}y+.  .  .=[«/] 


1  86  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  formulas  with  the  weight  symbols  introduced,  cor- 
responding to  those  found  in  the  preceding  articles,  are 
easily  derived  from  them  by  writing  a  \/~p  b  \/  T  .  .  .  /  |/J~ 
for  a,  /?,...  /,  and  a  \fy  /?  \/^  .  .  .  for  «,  flf  .  .  .  respec- 

tively.    (See  Art.  58.) 

Thus,  for  example,  from  the  normal  equations 

[paa\x  +  \_paV\y  +  .   .   .  =  [/«/] 

|>0>  +  \_pbb  -\y-\-  .   .   .  =  [>*/]  (2) 


we  should  have 

x  —  [«/]  =  [?/««][/>«/]  -|-  [>«/?][>£/]  +  .   .  • 


(3) 

and    by    equating   coefficients    of  /,,  /,,...    in    the    first 
expression, 

[uaa]al  -(-  [««/9]^,  -)-  .  •   •  =  «,«, 

(4) 


99.  To  Find  the  in.  s.  e.  //  of  a  Single  Observation. 

—If  the  errors  A  were  known  —  that  is,  if  the  n  observation 
equations  were 


where  XH,  ytt,  .   .   .   are  the  true  values  of  the  unknowns  —  we 
should  have  at  once 


But  we    have  only  the    residuals   i>   with    the    observation 
equations 


tf.rr  +  b,y  -f  .    .    .  -  /„  —  v,  (2) 


INDIRECT   OBSERVATIONS.  187 

where  x,  y,  .  .  .  are  the  most  probable  values  of  the  un- 
knowns. We  must,  therefore,  express  [JJJ  in  terms  of  the 
residuals  i>  in  order  to  find  it. 

From  the  two  sets  of  equations,  by  subtracting  in  pairs, 


Now,  taking  the  m.  s.  e.  /ix,  /Jty,  ...  to  be  the  errors  of 
A-,  y,  .  .  .  ,  that  is,  to  be  equal  to  xa  —  x,  y0  —  y,  .  .  .  ,  we 
have  from  Eq.  4,  Art.  93, 


x,  -  x 
and  therefore 


J,  =  V,  +  //(„,  4/[««J  +  /;.  Vtffi  +  .   .   .  ) 

Squaring,  adding,  and  attending  to  equations  10,  Art.  93,  we 
have  approximately,  ;/,-  being  the  number  of  unknowns, 

[^]  +  ;Wr  (4) 


Putting  [JJJ  =  ;//r,  there  results 

//-J^i-  (5) 

n  —  ;/,- 
the  expression  required. 

Reasoning  as  in  Peters'  formula,  Art.  47,  we  easily  de- 
duce from  (4) 

/'=  i.  2533-  (6) 

Vn(n—nf) 

which  is  known  as  Liiroth's  formula  (Astron.  Nadir.,  1740). 

When  »,-=  i,  equations  5  and  6  reduce  to  Bessel's  and 
Peters'  formulas  respectively  (Arts.  43,  47). 

100.  J\Iethods  of  Computing  [n>].  —  (a)  The  ordinary  method 
is  to  substitute  the  values  of  the  unknowns  found  from  the 
solution  of  the  normal  equations  in  the  observation  equa- 


188  THE   ADJUSTMENT   OF   OBSERVATIONS. 

tions,  and  thence  find  vlt  v»  .  .  .  The  sum  of  the  squares  of 
these  residuals  will  give  [vv~\. 

The  residuals  having-  to  be  found,  for  the  purpose  of 
testing  the  quality  of  the  work  this  method  of  computing 
[?'?>]  is  on  the  whole  as  short  as  any. 

As  checks  on  the  values  of  [_vv\  found  in  this  way  the 
following  are  of  value  : 

(b)  If  we  multiply   each   observation  equation  by  its  v 
and  take  the  sum  of  the  products,  then,  remembering  that 
[<77']  =  o,  [bv~\  =  o,  .  .   .,  we  find 

M=  -M 

(c)  If  we  multiply  each  of  the  observation  equations  by 
its  /  and  take  the  sum  of  the  products, 

•  -  -    -[//]  =  [*>/] 


(d)  We  have  for  two  unknowns,  x  and  y, 


=  \ad\x"  +  2\at].iy  +  [/;%*  -  2[al}x  - 

,   firi]         [rt/]Y  .  /..,,      [«* 
+  LJ/  -  L-j)  +  ([M]  - 


and  generally  for  m  unknowns 


INDIRECT    OBSERVATIONS. 


189 


Now,  from  (9),  Art.  86,  the  coefficients  of  [aa~], 
are  each  equal  to  zero.     Hence 


'  [££.l]         \_CC.2~] 

This  expression  was  first  given  by  Gauss  (De  Element is 
Ellipticis  Palladis,  Art.  13).  Its  form  suggests  that  if  we  add 
an  extra  column  to  the  normal  equations,  as  shown  in  the 
following  scheme,  we  shall  find  [z'z'J  at  the  same  time  as  the 
first  unknown.  This  is  analytically  very  elegant,  and,  as 
the  check  (see  Art.  87)  can  be  carried  with  this  column 
through  the  solution  of  the  normal  equations,  it  may  be 
used  for  finding  [?'?/],  if  one  is  computing  alone.  Only  one 
extra  term  [//]  has  to  be  computed  while  forming  the 
normal  equations. 

The  scheme  is  as  follows: 


X 

y 

z 

[aa] 

+   Fa6] 
*   [6*J 

:  m 

[an 

[6/1 

,  [a] 

\ff] 
[11} 

i 

[a6] 
[aa] 

far] 
[aa] 

£] 

.  [66.1] 

*  f6<Ml 

[W.I] 

+  [«.»] 

[//:!]  =  [//]  -  ("'][.«/] 

[aa] 

[*.i] 

[6/.I] 

i 

[W.i] 

[66.,] 

[«.,] 

Ir/2l                   rw  ii 

[^I,[//,]-^;-;[H,, 

1 90  THE   ADJUSTMENT   OF    OBSERVATIONS. 

Ex.  i.  To  find  the  m.  s.  e.  of  the  adjusted  values  of  the  unknowns  found 
in  Ex.  4,  Art.  83. 

The  first  step  is  to  find  [pvv].  This  we  shall  do  in  the  four  ways  in- 
dicated. 


(a) 


V 

P 

pvv 

—  0.05 

5 

O.OI 

—  0.36 

7 

0.91 

+  0.68 

4 

1.85 

—  0.03 

7 

O.OI 

—  0.62 

4 

r.  54 

4.32  =  [pvv] 

(b) 


p\l\v\  —  o 

pllrfJl  =  O 

pzlsv3  =  o 

pJtVi  =  7  X  0.76  X  —  0.03 
6/6z/5  =  4Xi.66X—  0.62 


—  o.  16 

—  4.12 


—  4.28  =  +  [pvl]  =  —  [pvv] 


(c) 


[pal~\x  =  —  5.32  X  —  0.05  =  0.27 
[pl>l~\y  -  -  6.64  X  —  0.36  =  2.39 
\_pcl~\z  =  +  11.96  X  +  0.68  =  8.13 


7  X  (o.76)2 
4  X  (i.66)2 


4-04 
=  11.02 


10.79 


15.06 
4.27    =  [pvv\ 


(d)    We  find  [///]  =  15.06. 


INDIRECT   OBSERVATIONS. 


The  solution  of  the  normal   equations,  with  the  extra  column  for  [///] 
added,  would  be,  according  to  the  foregoing  scheme, 


X 

y 

z 

12 

o 
+  II 

-  1 

-    4 
+  15 

-     5-32    1 
-    6.64 
+11.96 
+  15.06     >  2.36 

I 

o 

-   0.583 

-    0.443  J 

+  II 

-    4 
+  10.917 

—    6.64 
+     8.857 
+  12.70        (=15.06  —  2.36) 

I 

-    0.364 

-    o  .  604 

+    9.462 

+    6.422 
+    8.69      (=12.70—4.01) 

i 

+     0.68 

\pw]=     +    4.30       (=    8.69-4.39) 

Mean  value  of  [/ZT]  =  4.29. 
Hence  (Art.  99) 

and  (see  Ex.,  Art.  97) 


=, -.4, 


//*=  T.47  I'o.ii93,      //y=  i".47  t '0.1049,      /'«  =  i"-47  *  o.  1057 
=  o".5i  =  o".4S  =0^.48 


Ex.  2.  Show  that 

\_Av\  =  [vv] 

[Multiply  equations  i,  Art.  99,  by  vt,  v?,  .  .  .  and  add.     Then  since 

[av\  =  o,  [bv]  =  o,   .   .   . 
.•.[JiQ  =  -[/*]        ] 


192  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  3.  Show  that 


[Form  the  normal  equations  from  equations  3,  Art.  99,  and 

[aa](x0  -  x)  +  \ab}(y0  -y)  +  .  .   .  -  [a/J] 


since  [_av~\  =  [&v]  =  .  .  .  =  o 
Hence  from  Art.  100 


Ex.  4.   From  the  equation 

\al}x  +  \bl]y  +...-[//]=-  \vv] 


and 


deduce 


^AT.  5.   Prove  that  [<xz>]  =  [ySz/]  =  .   .   .  =  o 


Ex.  6.  From 


deduce  the  formula 


INDIRECT   OBSERVATIONS.  193 

1 01.  To  Find  the  Precision  of  any  Function  of 
the  Adjusted  Values  A'  K,  .  .  .—This  is  the  more  gen- 
eral  case  of  the  problem  just  discussed.  The  method  of 
solution  is  : 

First,  to  find  //,  the  m.  s.  e.  of  an  observation  of  weight 
unity,  and  next  pF  the  weight  of  the  function,  whence  the 
m.  s.  e.  ot  the  function  is  given  by 


where  nF  is  the  reciprocal  of/F 

The  value  of  p  is  computed  from  (5)  or  (6),  Art.  99. 
Next,  to  find  nF.     Let  the  function  be 

F=f(X,  F,  .   .  .  )  (i) 

in    which    A',  Y,  .  .   .    are    functions    of    the    independently 
observed  quantities  M^  Jlf.2,  .  .  .  KIn. 
By  differentiation 


and  therefore,  since  J/,,  Mv  .  .  .  are  independent, 


where  «„  ;/,,  .  .  .  Jtn  are  the  reciprocals  of  the  weights  of  the 
observed  quantities. 

(a)  Instead,  however,  of  using  this  general  formula 
directly,  it  is,  in  general,  a  great  saving  of  labor  to  compute 
from  a  modified  form  as  follows,  by  means  of  which  much 
of  the  work  already  done  in  solving  the  normal  equations 
may  be  utilized. 

Reducing  the  function  to  the  linear  form,  we  have, 
adopting  the  notation  of  Art.  81, 


IQ4  THE   ADJUSTMENT   OF   OBSERVATIONS. 

F=f  (*'  +  *,  Y'+y,  .  .  .  ) 


or,  as  it  may  be  written, 

dF=Gs+G.y+...  (3) 

Now,  since   ,r,  y,  .  .  .  are    not   independent,  but  are  con- 
nected by  the  equations 

[*«]*+  [«*b+.   •  •  =  [«/] 


we  must  get  rid  of  this  entanglement  by  expressing  these 
quantities  x,  y,  ...  in  terms  of  /,,  /2,  .  .  .  ,  which  are  inde- 
pendent of  each  other.  From  Art.  93  we  may  write 


where  «,,  «s,  .  .  .  ;  /31,/32,  ...;...  are  functions  of  <7,,  b^  .  .  .  ; 
a^  bv  ...;...      Hence,  substituting  in  (3) 


and,  therefore  (Arts/  19,  56),  since  the  observation  equations 
have  been  reduced  to  weight  unity, 


[/9,9]  +  .  .  .  (4) 


where  [««],  [«;9]  .  .   .  may  be  found  in  the  manner  indicated 
in  Arts.  94,  96,  or  97.      Hence  £/is  known. 
(b)  Eq.  4  may  be  written 

«F=G&  +  G&+.  .  .  +G.Qn  (5) 


INDIRECT   OBSERVATIONS. 

w  here 

a  =  [«*]£,  + 


that  is,  where  (see  Eq.  i,  Art.  97) 

+•  •  • 

.  •  •  (7) 


Hence  the  weight  of  a  function 

G>+6^+.  .  . 
of  several  independent  unknowns  x,  y,  ...  is  found  from 

»jr=[£<2] 

where  (2,,  (22,  .   •  •  satisfy  the  equations 


Therefore  we  conclude  that,  z/~  /«  #  scries  of  observation 
equations  the  values  of  the  unknowns  x,  }>,  .  .  .  are  required,  as 
well  as  tJieir  weights  or  the  weight  of  any  function  of  them, 
these  results  can  be  found  at  one  time  by  making  a  solution  of 
the  normal  equations  for  finding  .v,  }',...  in  general  terms,  and 
then  substituting  for  [W],  [/>/],  •  •  •  their  numerical  values  on 
the  one  hand  and  the  values  of  G^  Gz,  .  .  .  on  the  other. 

(c)  This  result  may  be  stated  in  other  forms.  Thus  from 
Eq.  4,  by  substituting  for  [««],  [«^],  .  .  .  their  values  from 
Art.  96,  or  by  substituting  for  (9,,  Q.,,  .  .  .  their  values  in  (6) 
as  expressed  in  Art.  96,  we  have,  alter  a  simple  reduction, 

G?_      (G1RA±  G^      (C.fl.+  C.S.  +  g.)' 
Uf~[aa^         [M.i]  \cc.2\ 

Comparing  this  expression  with  (d),  Art.  100,  it  is  evident 
that  the  several  terms  are  such  as  would  result  from   the 
26 


196 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


following  elimination  (Ex.  three  unknowns)  by  finding  the 
products  of  the  quantities  bracketed  : 


M 

M 

M 


[«*] 


[fc.1] 


[aa\ 
G-jA',  + 


(d)  The  expression  (8)  for  iip  may  be  easily  transformed 


into 


where 


Circumstances  must  decide  which  of  the  four  forms  given 
should  be  chosen  in  any  special  case.  A  machine  can  be 
used  to  the  best  advantage  with  the  second  and  third  forms. 
The  third  form  is  also  convenient  when  the  weights  alone 
are  required,  without  the  values  of  the  unknowns,  and  the 
second  when  the  values  of  the  unknowns  can  be  found  by 
an  easier  method  of  solution  than  the  Gaussian  method  of 
substitution. 


INDIRECT   OBSERVATIONS. 


I97 


Ex.  i.  In  Ex.  4,  Art.  83,  it  is  required   to  find   the  m.  s.  e.  of  the  angle 
PSB. 

The  function  is 

dF=  -x  +  z 

First  Solution.  From  Eq.  4, 

up  -  [aa]  -  2[ay]  +  [yy] 

=  0.1193  —  2  X  0.0616  4-  0.1057  (from  Ex.,  Art.  97). 
=  o.  1018 

Second  Solution.   From  equations  7, 


Hence 
and 


-    4(?3=      o 


(?,=  —0.0577,  (?3  =  +  0.0440 

«p=  -  i  X  —  0.0577  +  i  X  0.0440 
=  0.1017 


Third  Solution.  Add  the  extra  column  G  to  the  solution  of  the  normal 
equations,  which  would  give  the  scheme 


X 

y 

z 

G 

+  12 

-  7 

—  i 

+  ii 

—  4 

o 

+  15 

+  i 
-0.0833 

+   I 

-0.5833 

+  ii 

—  4- 
+  10.9169 

0 

+  0.4169 

0 
+  0.4169  - 

+  0.0441 

1 
1 

\ 

+  i 

—  0.3636 
+  9-4625 

+  i 

Hence 


UP—  -  ix  —  0.0833  +  0.4169  X  0.0441 
—  0.1017 


198  THE   ADJUSTMENT  OF   OBSERVATIONS. 

Fourth  Solution. 

I2/£0  =:  +  I 

II/&!  =          O 


.'.   k0=  0.0833,  ^1=0,  k-i—   —0.0440 

up  —  (0.0833)-  X  12  +  (0.0440)^  X  9-4625 

—  o.  1016 
Also 

j.ip°=  i"-47  Vo.102 


Ex.  2.   Given  the  observation  equations 

diX  +  b^y  =  A 
a?x  +  b?y  =  I?. 

anx  +  bny  —  /« 

to  find  the  weight  of/Jc  +  gy. 
[The  normal  equations  are 

\aa~\x  +  [ali\y  =  [>/] 


1  02.  To  find  the  average  value  of  the  ratio  of  the  weight  of 
the  observed  value  of  a  quantity  to  that  of  its  adjusted  value  in 
a  system  of  independently  observed  quantities. 

The  adjusted  value  of  the  first  observed  quantity  Ml  is 
Ml  -f  v,.  From  Art.  Si  it  follows  that  the  weight  of  M1  -f  v, 
is  the  same  as  the  weight  of  /,  -\-  vt.  Now, 

Ii+vl  =  a1*  +  t>ty+  •   •   •  (0 

Hence  if  Pl  is  the  weight  of  the  adjusted  value  Jlft  -\-v^  that 
is,  is  the  weight  of  the  function  a^x  -f-  l^y  -f-  .  .  .,  and 
pvpv  .  .  .  are  the  weights  of  /1?  lv  .  .  .,  we  have 


where  (see  Eq.  4,  Art.  98,  (b)  Art.  101) 


Ql  — 

Q,  =  [>«£k  +  [«^]^  +  .   .   .  -  «  A  (3) 


INDIRECT   OBSERVATIONS.  199 

Therefore  by  substitution  of  Qlt  (?2,  .   .   .   in  (ty, 

J5  =  «,«,  +  &jt  +  -   -   - 
Similarly 


Hence  by  addition 

]    =  M  +  [4?]  +  ...   to  ;/,.  terms 

=  ;/,-,  the  number  of  independent  unknowns. 

This   result    may   be    very    readily   derived    directly  as 
follows:   In  (i)  put  [«/]  for  x,  [,?/]  for/,  .   .   .  ,  and  we   have 


Hence,  since  /,,  /„,...   are  independent, 


as  before. 

£JT.   To  check  the  weights  of  the  adjusted    values  uf  the   angles  found  in 
Ex.  4,  Art.  83. 

The  weights  of  the  measured  values  of  the  angles  are 

5,    7-    4,    7,    4 
The  weights  of  the  adjusted  values  are  (Ex.,  Art.  <)5  ;  Ex.  i,  Art.  101) 

8.4,    9.5,    9.5,    9.8,    7.5 
Also 

^+-  +  -  +  ^  +  - 
b.4      9.5      9.5      9.h      7.5 

=  3 

=  the  number  of  independent  unknowns, 
as  it  should. 


200  THE  ADJUSTMENT   OF   OBSERVATIONS. 

103.  From  the  principle  just  proved  we  may  derive  a 
proof  of  the  formula  found  in  Art.  99  for  the  m.  s.  e.  of  an 
observation  of  weight  unity  in  a  series  of  n  observations  in- 
volving fif  independent  unknowns.  It  is  analogous  to  the 
proof  given  for  the  m.  s.  e.  of  a  single  observation  in  a  series 
of  n  observations  of  one  unknown. 

The  errors  of  the  observed  values  M1}  Mv  .  .  .  MH  are 
Ju  J2,  .  .  .  4,,  and  the  errors  of  the  most  probable  values 
Vv  Vv  .  .  .  Vn  may  be  assumed  to  be  //,-,,  /vs,  .  .  .  pVn  re- 
spectively. Hence  since 

M=  V-v 
we  have 

4  =  t^i  —  vi 
=      -  v 


and,  therefore, 

[/  J  J]  =  [frwr]  +  [/  mi\  (  I  ) 


Again,  since  P1  is  the  weight  of 


where  jj>  is  the   m.  s.  e.  of  an  observation  of  weight  unity. 
Hence 

A/V-4"2 

* 


tnpv*=  —  / 

*  i    *n  --    r>  ' 

*H 

By  addition, 


(2) 


INDIRECT   OBSERVATIONS.  2OI 

But  by  definition 


11 
Substituting  in  (i),  we  have  finally 

I?  —  \-Pvv\ 

71  —  tti 

the  result  required. 

Miscellaneous  Examples  and  Artifices  of  Elimination. 

In  this  section  will  be  discussed  several  problems  of  im- 
portance, and  also  certain  artifices  of  elimination  which  may 
often  be  employed  with  advantage  in  the  solution  of  obser- 
vation equations. 

104.  The  labor  of  solving  and  finding  the  values  of  the 
unknowns  may  often  be  shortened  by  taking  advantage  of 
some  principle  inherent  in  the  form  of  the  observation 
equations  themselves.  For  example,  if  we  have  a  series  of 
observation  equations  containing  two  unknowns,  and  of 
which  the  coefficient  of  the  first  unknown  is  unity,  instead 
of  solving  in  the  usual  way  we  may  reduce  the  observation 
equations  to  equations  containing  the  second  unknown  only, 
and  thus  solve  more  readily. 

Given 

*+^J'  =  A     weight/, 

*  +  t>*y  =  t,        "     A 

Forming  the  normal  equations  in  the  usual  way,  we  have 


whence  eliminating  x, 


Now,  if  the  first  normal  equation  is  divided  by  [/>],  so  that 

,    Mr  _[ 

~' 


2O2  THE   ADJUSTMENT   OF   OBSERVATIONS. 

and  from  this  equation  each  of  the  observation  equations  in 
succession  is  subtracted,  there  result  the  equations 


The  normal  equation  for  finding  y  from  these  equations  is 


the  same  as  results  from  the  elimination  of  x  in  the  normal 
equations. 

This  process  is  specially  convenient  if  the  original  ob- 
servation equations  are  numerous  and  the  coefficients  l\, 
bv  .  .  .  and  the  terms  tlt  /„,...  large  and  not  widely  dif- 
ferent. 

Ex.  To  solve  the  equations  in  Ex.  3,  Art.  83. 
The  mean  of  the  equations  is 

*-  0.97.7  =  -  1.97 
Subtract  each  equation  from  this  mean  equation,  and 

+  O.62.7  =    +  O.OI 

+  0.477  =  -0.07 
+  0.267  =  +  0.03 

—  0.017  =  —  0.02 

—  0.257=  +  o-°3 

—  i.o8ji'=       o.oo 

The  normal  equation  formed  from  these  equations  is 

+  i.9U'=  —  0.27 
and  .  '  .         y=  —  ox.oi4 

Substitute  for^j'  its  value  in  the  mean  equation,  and 


105.  Again,  we  may  take  advantage  of  the  presence  in 
the  problem  of  some  arbitrary  quantity  to  which  a  con- 
venient value  may  be  assigned.  Thus,  to  find  the  difference 
of  the  coefficients  of  expansion  of  two  standards  A  and  B 


INDIRECT   OBSERVATIONS.  203 

from  observed  differences  of  length  at  certain  fixed    tem- 
peratures. 

Let  x        -  the  excess  in  length  of  A  over  B  at  an  arbitrary 

temperature  fa. 
y  =  the  excess  of  the  coefficient  of  expansion  of  A 

over  that  of  B, 
/,,  /„,...  =  the  observed  differences  in  length  at  tempera- 

tures /1(  /2,  .  .  .  and  whose  weights  are  /,,  />.,,  .  .  . 

We  have  then  the  observation  equations 

-r  +  (',  -  t0)j'  -  /,  =  <  -,     weight  A 

x  +  (/a  -  /„)  y  -  L  =  7-a     weight  A  (  i  ) 

and  the  normal  equations 


<»Y)  y     =  \.C  -  0//1      (2) 

As  the  value  of  /„  is  arbitrary,  the  normal  equations  will 
be  simplified  by  taking  it  equal  to  the  weighted  mean  of 
the  temperatures  ;  that  is, 


"AJ 
and  they  will  then  become 


O'lJ'  -  [('  -  OXJ  (4) 

from  which  .i'  and  y  are  found  at  once,  with  their  weights  at 
the  mean  temperature  /„. 

If  the  values  of  /are  numerically  large  it  will  lessen  the 
labor  of  finding  the  value  of  y  if  the  mean  value  of  .r  found 
from 


is  substituted   in  the  observation  equations  before  the  nor- 
mal equation  in  y  is  formed.      We  should  then  have 


from  which  to  find  j. 


2O4  THE   ADJUSTMENT   OF   OBSERVATIONS. 

It  is  evident  that  the  value  of  y  found  in  this  way  is  the 
same  as  before.     For 

OC  -  '„)('-  *)]  =  EX*  -  '.)']  - 


since  the  coefficient  of  x  is  equal  to  zero  from  Eq.  3. 

The  quantity  I—  x  comes  in  very  conveniently  in  com- 
puting the  residuals  v  in  finding  the  precision. 

The  precision.  —  If  n  is  the  number  of  observations,  the 
number  of  unknowns  being  2,  we  have  for  the  m.  s.  e.  //  of  a 
single  observation 


and 


The  length  at  any  temperature  /'  is 

*  +  (t-*t.) 

and  its  m.  s.  e.  //.,.,  is  found  from 


The  weight  is  greatest  when  /ix.  is  least,  that  is,  when 


Ex.  The  following  were  among  the  observations  made  for  the  determina- 
tion of  the  difference  of  length  between  the  Lake  Survey  Standard  Bar  and 
Yard  ;  and  also  for  the  difference  between  their  coefficients  of  expansion. 
The  unit  is  Tn7?fl?f7r  inch. 


INDIRECT    OBSERVATIONS. 


205 


Required   the  difference  of  length  at  62°  Fahr.  and  at  any  other  tempera- 
ture /. 


Date. 

Observed  temp.  (/). 

Bar-  Yard  (/). 

Weight  </>). 

1872,  March  5 

24°.  7 

791 

5 

"      M 

37°-i 

SlI 

I 

"     26 

6i°.7 

833 

6 

April    4 

49°-3 

820 

6 

"       12 

66°.  8 

847 

8 

"       20 

7i°.  5 

849 

8 

Let  x  =  the  most  probable  difference  between  Bar  and  Yard  at  62^  Fahr., 
y  =  the  most  probable  difference  between  coefficients  of  expansion  of  Bar 

and  Yard. 
The  observation  equations  will  be  of  the  form 

x  +  (t  —  62)y  —  I  =  v 
The  computation  is  arranged  in  tabular  form  as  follows  : 


p 

ft 

pi 

t-t» 

l-x 

5 

123.5 

3955 

-  32.2 

—  40 

i 

37-1 

Sn 

-  19.8 

—  20 

6 

370.2 

499s 

4-  4-8 

+   2 

6 

295.8 

4920 

-  7-6 

—  II 

8 

534-4 

6776 

+  9-9 

4-  16 

8 

572.0 

6792 

+  14.6 

+  18 

34 

1933.0 

28252 

>«  =  56  .9 

x  —  831 

P(t-ttf  ' 

/(/-/„)(/-*) 

y(t  -  /«) 

V 

pw 

5184.20 

6440.0 

—  40.6 

—  0.6 

i.S 

392.04 

396.0 

-  24.9 

-4-9 

24.0 

138-24 

57.6 

+  6.0 

+  4.0 

96  .  o 

34<>-5('> 

501.6 

-  9-6 

+  1.4 

II  .8 

784.08 

1267.2 

+  12.5 

-3-? 

98  .  o 

1705.28 

2102.4        +  1^-4 

+  0.4 

i  .3 

8550.40 

10764.8 

232.9 

y  =  1.26 

2C>6  THE   ADJUSTMENT    OF   OBSERVATIONS. 


7232.9 

=V  6^2  = 
7-6 


7.6 


\  8550.40 

Value  of  .r  =  831      at  56°.  9 
6.4          5°-  1 


=  837.4       62°.  o 
//ea«  =  (i.3)»+  (5.  i)2  X  (o.i)2  =  1.9 

These  values  may  be  checked  by  computing  by  the  ordinary  process. 
The  application  of  the  preceding  method  to  the  solution  of  this  problem 
is  due  to  Mr.  E.  S.  Wheeler. 

106.  It  often  happens  that  of  two  series  of  observed 
quantities  one  is  a  function  of  the  other,  and  that,  therefore, 
the  observed  quantities  may  be  regarded  as  co-ordinates  of 
points  in  the  curve  which  represents  the  relation  connect- 
ing them.  From  the  plot  of  the  points  the  general  form  of 
the  curve  can  be  judged,  and  then  the  special  form  which 
satisfies  the  observations  may  be  found  as  follows. 

Let  us  investigate  the  important  practical  case  where  the 
co-ordinates  have  been  equally  well  measured  and  the  curve 
is  a  straight  line,  both  co-ordinates  being  regarded  fallible. 

Let  x^y^  :  x»  y^  :  .  .  .  xn,  yn  be  the  observed  values  of 
the  co-ordinates,  and  X^  F,  :  Xv  F2  :  .  .  .  Xn,  YH  their  most 
probable  values. 

The  conditions  to  be  satisfied  are: 


(i) 


where  a  and  b  are  constants,  to  be  found   from  the  observa- 
tions.    Also 

(Xt  -  *,)'  +  (  F,  -;0*  +  ...  -  a  min.  (2) 

Eliminate   the  F's  by  substituting  from  equations   i    in  (2), 
and  then 

(X,  —  x^f  -J-  (aXl  -f-  b  —  }\}y  -(-  .  .   .  =  a  minimum. 


INDIRECT   OBSERVATIONS. 


207 


Differentiate   this  equation  with  respect  to  the  independent 
variables  Xlt  X.,,  .   .  .  XH,  and 


X,  —  ,ra  -f  a(aX,  -f  b 


=  o 


from  which  equations  Xlt  X.2t  .  .  .  XH   may  be  expressed  in 
terms  of  a,  b  and  the  known  quantities  ,\\,  }\  ;  .   .   .  ;  xn,  yn. 
Again,  differentiate  with  respect  to  a,  b,  and  we  find 


Hence  a  and  /;  are  known,  and  therefore  the  equation  of  the 
straight  line  is  known. 

The  same  mode  of  reasoning  may  be  applied  to  the  pass- 
ing of  a  curve  of  a  specified  form  through  n  points,  whether 
their  co-ordinates  have  been  equally  well  measured  or  not. 

The  problem  of  passing  a  line  which  should  deviate  as 
little  as  possible  from  the  positions  of  several  given  points 
was  discussed  by  Lambert*  as  long  ago  as  1765,  thirty 
years  before  Gauss  first  made  use  of  the  method  of  least 
squares.  Lambert  reasoned  that  if  the  line  is  supposed 
drawn  the  points  should  deviate  as  much  on  the  one  side  as 
on  the  other,  and  hence  that  the  line  would  pass  through 
the  centre  of  gravity  of  the  points. 

107.  Potheiiot's  Problem. — In  topographical  work  the 
three-point  problem,  usually    called 
Fothenot's  problem,  is  of  great  im- 
portance.    It  may  be  stated  as  fol- 
lows : 

Let  Plt  P..,  P3  be  three  points  whose 
positions  are  known- — as,  for  example, 
by  their  co-ordinates  with  reference  to 
known  axes — and  P  a  point  at  which 
the  angles  P^PP.,  =  c, :  PJ'P.,  =  cr,  are  o 


Fig. 10 


*  Jicylragc  ~u»i  Gclo'auchc  dcr  Matlieiimtik.     Berlin,  1765. 


2O8  THE   ADJUSTMENT   OF   OBSERVATIONS. 

observed.  It  is  required  from  these  data  to  determine  the 
position  of  P. 

The  geometrical  solution  is  simple.  Plot  the  points 
Plt  PV  Py  Describe  on  the  line  1\P^  a  circle  containing  an 
angle  equal  to  <plt  and  on  P^P3  a  circle  containing  an  angle 
equal  to  <pz.  The  intersection  of  these  circles  will  be  the 
point  required. 

A  solution  much  used  in  practice — as,  for  example,  in 
plotting  soundings — is  to  lay  off  on  a  piece  of  tracing-linen 
two  adjacent  angles  equal  to  <f>1}  yv  and  then  move  this 
figure  over  the  map  on  which  the  points  1\,  P»,  Pa  are 
plotted  until  the  directions  PPlt  PPV  PP3  lie  over  the  points 
Pn  Pv  P3  respectively.  The  position  of  P  is  then  pricked 
through. 

So  far  as  the  plotting  of  soundings  is  concerned,  the  above 
is  in  general  sufficient;  but  in  the  location  of  important 
secondary  points  with  reference  to  known  primary  points 
greater  precision  is  necessary. 

The  position  of  P  may  be  computed  trigonometrically 
as  follows:  Since  the  positions  of  the  points  Flt  Pv  P3  are 
known,  the  lengths  of  the  lines  1\P»  P^P^  PJ\  ^re  known, 
and,  therefore,  the  angle  P^P*  can  be  found.  Call  its 
value  ft. 

Denote  the  angles  PP^PV  PPSP~2  by  «j,  «2  respectively. 

The  sum  of  the  angles  of  each  of  the  triangles  PPtP3, 
bein  180°,  we  have 


and,  therefore, 

«,  +  «•.<  =  36o°  -  ?,  -  f  „  -  /? 
a  known  quantity. 
Again, 

PP,         />./»,  PPn_         P^_ 

sin  «j       sin  ^,  sin  «.,       sin  ^ 

sin  «2  _  P^  sin  ^2 


. 
sin  a 


..,  ,, 

'   "    .  -  -  —  tan  y  suppose, 
PJ    sin   > 


3 


INDIRECT   OBSERVATIONS.  209 

and 

sin  a1  -\-  sin  «a  _  I  -f-  tan  y 
sin  a,  —  sin  «,^       i  —  tan  y 
or 

cot  I  (a,  -  «,)  =  tan  (y  +  45)  cot  I  (a,  -f  «2) 

and,  therefore,  «,  —  «,  is  known.  Combining-  with  the  value 
of  «,  -(-  «.2,  we  find  «,  and  «.,. 

Mence  the  position  of  Pis  completely  determined. 

It  often  happens,  as  in  the  determination  of  the  position 
of  a  light-house,  for  example,  that  more  than  three  points 
are  sighted  at  from  the  point  occupied,  and  that  there  are 
more  observations  than  are  necessary  to  locate  the  point. 
Its  most  probable  position  can  consequently  be  found.  We 
proceed  to  show  the  method  of  finding  it. 

Let  Plt  P.2,  .  .  .  be  the  known  points  in  order  of  azi- 
muth, as  seen  from  the  unknown  point  P.  X^  Yl  :  X^  Y.2  ;  .  .  . 
the  co-ordinates  of/3,,  /*„...  referred  to  some  known  sys- 
tem of  rectangular  axes,  preferably  the  parallel  and  meridian 
at  the  point  chosen  as  origin  of  co-ordinates.  0,,  0,,  .  .  . 
the  angles  which  the  most  probable  positions  of/3/",,  PP.2,  ,  .  . 
make  with  the  axis  of  x.  $,',  #/,  .  .  .  approximate  values 
of  these  angles  as  computed  from  the  co-ordinates,  and 
#„  #,,  .  .  .  corrections  to  these  approximate  values,  so  that 


y,,  (£„  .  .  .  the  angles  observed  at  P  between  the  directions 
/•Y',  PP.,  .  .  .  and  the  initial  direction  PI\. 

X,  Y,  .  .  .  the  co-ordinates  of  P  referred  to  the  same  axes 
as  A'lt  Yl  :  A\,  Y.,;  .  .  .  Suppose  X',  Y'  to  be  close 
approximations  to  the  values  of  X,  F,  found  graphically, 
and  ,r,  y  corrections  to  these  values,  so  that 

X=zX'  +  xt  Y=  Y'+y, 

Now, 

Y  -  -  F, 
tan  0  = 


2IO  THE   ADJUSTMENT   OF   OBSERVATIONS. 

or 

V—  Y  A-  v 
tan(0/  +  *0  =  £*d 

A     -  A!  -\-  * 

Take  logs,  of  both  members  and  expand,  then 
log  tan  0/  +  dfr  =  log  ( F '  -  F,)  +  o j,  -  log  (A-  -  A',)  -  o 

where  o,,  o2,  o3  are  tabular  differences,  as  explained  in  Art.  7. 
But 


F'  —  F 

tan  #/  —  -T77 ^r1,  and  is,  therefore,  known. 

A    —  A, 


Hence 


j  suppose, 


or 

Similarly 


0=  *' 


Now,  comparing  computed    and    observed    values   of  the 
angle  P,P1\, 


Calling  the  quantity 

-  ft  -  (^/  -  0/)  =  A 

the  observation  equation  may  be  written 

(a,  -  a^x  +  (l\  -  b^y  =  /, 

An  observation  equation  for  each  of  the  other  angles 
yv  <f>3,  .  .  .  observed  at  P  may  be  formed  in  the  same  way. 

In  the  expansions  above  we  have  used  log.  differences 
as  being  most  convenient  and  as  leading  to  results  close 
enough  in  problems  of  this  kind.  Five-place  tables  are 
sufficient;  indeed,  in  most  cases  four-place  are  ample. 


INDIRECT  OBSERVATIONS.  211 

On  this  problem  consult  Bessel,  Zach's  Monatliche  Cor- 
respondent, vol.  xxvii.  p.  222;  Gerling,  Pothenotsche  Aufgabe, 
Marburg,  1840  ;  Gauss,  Astron.  Nachr.,  vol.  i.  No.  6  ;  Schott, 
C.  S.  Report,  1864,  App.  13;  Petzold,  Zeitschr.  fiir  Vermess., 
vol.  xii.  p.  227. 

Ex.  Given  the  rectangular  co-ordinates  of  six  known  points,  and  the 
angles  observed  at  the  point  P  whose  position  is  to  be  determined,  as  follows 
(C.  S.  Report,  1864,  App.  13): 

X,  =  1845.0       F,  =  5534.0  cpt  =    61    12'  10" 


X-,  —  1485.0 

F,  =  2486.7 

cp.,  =  97°  48'  27' 

X3  =   o.o 

F3  =   o.o 

<Pa  =  195°  55'  56' 

JT4  =  4418.2 

F4  =  1416.7 

<pi  =  205°  35'  04 

^  =  6163.8 

F5=  398.1 

ep5  =  215°  53'  44 

Xr,  =  5810.6 

F6=  1255.7 

to  find  the  co-ordinates  of/3. 

By  plotting  the  points  we  may  take  off  the  approximate  values 

X'  =  344S        Y'  =  2440 

Following  the  form  laid  down  in  the  preceding  Art.,  and  using  five-place 
log.  tables  (Art.  7), 

log  (3094  —  r)      =  3-49052    —  14-0;' 
log  (—  J6o3  —  x)=  3.20493;*  +  27-ur 


.'.  log  tan  6V  +  <5i$i  =  0.28559;?  —  27.  ix  —  14.01- 

Now,  fc>/  =  117'  23'  19"  and  5,  =  —  31  for  i' 

Hence 

<r)i  =  117°  23'  19"  +  52.5^  +     27.1;' 

Similarly                        6>2  =  178°  38'  14"  +  2.5_r  +  103.  Gy 

W3  =  215°  17'  07"  —  28.0*  4-    39.  6j 

t)4  =  313°   28'  28'  —  IO3.  2.V  —     97-8^ 

65  =  323"  03'  45"  —    35.6.V-    47.  3j 
^f.  =  333"  22'  37"  -    35.6.V  —    71.0;' 

Subtracting  <->i  from  each  of  the  values  W2,  (-)3,  ...  in  succession,  and  com- 
paring with  the  measured  values  <p,,  <pa,  .  .  .,  we  have  the  observation 
equations 

50.  ar  —    76.5^'  =  165 

So.5.v—    1  2.  5^=32  1 

I55-7-*-  +  124.97=:  553 

SS.i.v+     74.47=322 

83.IJT4-    98.1  v  —  334 

28 


212  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Hence  the  normal  equations 

48746*  +  298 13  y  =  177986 
29813*  +  36767^=  109157 

The  solution  gives 

x  =  +  3.66     weight  24572 
y=—  0.02    weight  18533 

which  added  to  the  approximate  values  above  give  the  final  values 

X=  3451. 66       'Y=  2439. 98 
Substitute  for  x,  y  in  the  observation  equations,  and  we  have  the  residuals 

Vi  =  +  19.5,  z/2  =  —  27.8,  va  —  +  8.1,  v4  =  +  i.o,  7/5  =  —  13. 5 

Hence,  since  \vv\  =  1402, 

,.  _  .  /  1402 

f*  —  4/  — =  22 

Y    5-2 

22 

and  JLlx== —  —  ai4 

^24572 

22 

HY—  =  0.16 

^18533 


CHAPTER  V. 

ADJUSTMENT    OF    CONDITION    OBSERVATIONS. 

108.  We  now  take  up  the  third  division  of  the  subject  as 
laid  down  in  Art.  39.  So  far  the  quantities  we  have  dealt 
with,  whether  directly  observed  or  functions  of  the  quan- 
tities observed,  have  been  independent  of  one  another  ;  but 
if  they  are  not  independent  of  one  another  —  that  is,  if  they 
must  satisfy  exactly  certain  relations  that  exist  a  priori  and 
are  entirely  separate  from  any  relations  demanded  by  ob- 
servation —  they  are  said  to  be  conditioned  by  these  relations. 

All  problems  relating  to  condition  observations  may  be 
solved  by  the  rules  laid  down  in  the  preceding  chapters. 

Let,  with  the  usual  notation,  V»  V»  .  .  .  Vn  denote  the 
most  probable  values  of  n  directly  observed  quantities 
Mlf  Mv  .  .  .  Mn  whose  weights  are  A»  A?  •  •  •  P»  respectively. 
Let  the  nc  conditions  to  be  satisfied  exactly  by  the  most 
probable  values,  when  expressed  by  equations  reduced  to 
the  linear  form,  be 

a'V       a"l         .   .   .  -L'  =o 


where  a',  a",  .  .  .  ;  b'  ,  />",  .../...;  L'  ,  L  ',  .   .   .  are  known 
constants. 

If  v^  vv  .   .   .  vn  denote  the  most  probable  corrections  to 
the  observed  values,  so  that 

V^-M^v, 
V,  -  M,  =  v, 

....  (2) 


214  THE   ADJUSTMENT   OF   OBSERVATIONS. 

we  have  the  reduced  condition  equations 

a'v^  -f-  a"v^  -f-  .   .   .  —  /'  =:  o 
b'v,  -f  £'V2  +  .   .   .  -  I"  =  o 

or  [av\  —  I'  =  o 

L       J  V.J/ 

where  I'  =  L'  -  \_aM~\,  I"  =  L"  -  \bM\  .  .    .,  and  are,  there- 
fore, known  quantities. 

The  most  probable  system  of  corrections  is  that  which 
makes 

\_pvv~\  =  a  minimum,  GO  suppose. 

The  problem  is  to  solve  this   minimum  function  when  the 
corrections  v  are  subject  to  the  above  nc  conditions. 

Direct  Solution — Method  of  Independent   Unknowns. 

109.  It  is  plain  that  nc  of  the  corrections  can,  by  means 
of  the  condition  equations,  be  expressed  in  terms  of  the 
remaining  n  —  nc  corrections,  and  that  by  substituting  these 
nc  values  in  the  minimum  function  we  should  have  a 
reduced  minimum  function  containing  n  —  nc  independent 
unknowns.  This  function  can  be  found  in  the  usual  way 
by  equating  to  zero  its  differential  coefficients  with  respect 
to  each  unknown  in  succession.  The  n  —  nc  resulting  equa- 
tions, taken  in  connection  with  the  nc  condition  equations, 
determine  the  n  corrections  i\,  i\,  .  .  .  vn.  Thence  \_pvv~] 
is  found. 

The  solution  of  the  n  —  nc  equations  can  be  carried 
through  by  any  of  the  methods  of  Chapter  IV.  The  pre- 
cision of  the  adjusted  values,  or  of  any  function  of  them, 
can  also  be  found  as  in  Chapter  IV. 


CONDITION   OBSERVATIONS.  21  5 

Ex.  i.  Take  that  already  solved  in  Ex.  4,  Art.  83. 

Let  Vi,  vi,  v3,  v^,  z>»  be  the  most   probable  corrections  to  the  measured 
angles,  then  the  conditions  to  be  satisfied  are 

PSB  +  z/4  =  FSB  +  v*  —  FSP  —  vi 
OSB  4-  z>5  =  FSB  +  v3  —  FSO  —  vi 

Substituting  for  PSB,  FSB,  etc.,  their  measured  values,  the  condition  equa- 
tions may  be  written 

v\  —  v3  +  vt  =  —  0.76 
Vi  —  v3  +  z/s  =  —  1.66 
wiih 

$v  i~  +  7V*1  +  4^32  +  7Vt-  +  4rv  =  a  min. 

Substitute  for  z/4,  ?'B  in  the  minimum  equation,  and 

SVia-  +  7~v  +  4^a2  +  7(z'i  —  Va  +  0.76)2  +  4(1/2  —  v3  +  i.66)s  =  a  min. 


Hence,  differentiating  with  respect  to   TI,  r%  v3  as  independent  variables,  we 
have  the  normal  equations 

I2-c',  -       7<'3  =    —       5.32 

iir-a—    47'3  =    -    6.64 

—   7^1—       4c/2   +    157-3   =  11.96 

whence 


and  from  the  condition  equations 

T'4  =  —  0".03       I/8  =—  0".62 

These  results  are  the  same  as  those  already  found  on  p.  149. 

Ex.  2.  The   angles   A,    J5,  C  of  a    spherical    triangle    are    equally     well 
measured  ;  required  the  adjusted  values  and  their  weights. 
The  condition  equation  to  be  satisfied  is 

A  +B  +  C-  180  +  e  (i) 

where  e  is  the  spherical  excess  of  the  triangle. 

Putting  yl/,  +  :•,,  J/3  +  -'„,  M3  +  7-3  for  A,  B,  C,  the    condition    equation 
becomes 

''i  +  ''i  +  r-3  =  1  80  +  £  —  [,V] 

=    /  suppose  (2) 

Also 

"V  +  7'2!  +  TV  =  a  min. 

Substitute  for  7'3  from  (2)  in  the  minimum  function,  and 

7V  +  7'3S  +  (?•,  +  7',  —  /)3  =  a  min. 


2l6  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Differentiating  with  respect  to  the  independent  variables  vi,  z>a,  and 


V\  +  2V*  =  l  (3) 

which  give 


Also  from  Eq.  2, 


Hence  the  correction  to  each  angle  is  one-third  of  the  difference  of  the  theoretical 
and  measured  sums  of  the  three  angles. 

To  find  the  weight  of  the  adjusted  value  of  an  angle,  as  A. 

The  function  is 

dF  '  =  v\ 

Hence,  following  the  method  of  Art.  101  (b), 


where  G\  =  i,  and  Qi,  @2  are  found  from 

2(?i   +      (?2  =  I 

@i  +  2<2-,  =  o 
that  is,  weight  of  A  =  f  if  weight  of  measured  value  is  unity. 

Check.  Weight  of  direct  measure  of  A  =  i 

Wt.  of  indirect  meas.  (=180+  £  —  B  —  C)  of  A  =  % 

Weight  of  mean  =  i 
as  already  found. 


Ex.  3.  To  find   the  weight  of  a  side,  a,  in  a  triangle  all  of  whose  angles 
have  been  equally  well  measured,  the  base,  b,  being  free  from  error. 

sin  A 
Here  F-=a  =  b—r- 

sin  B 

.'.  dF  —  a  sin  i"  (cot  A  v±  —cot£  r-j) 
The  weight  is  found  from 

uf  =  a  sin  i"  cot  A  Qi  —  a  sin  i"  cot  B  Q2 

where  Qi,  Qi  satisfy  the  equations  (Art.  101) 

2Qi  +    (?a  —      a  s'n  T    cot  -^ 
Qi  +  "2.Q.I.  =  —  a  sin  i"  cot  B 
Hence 

Up  —  ^a''  sin2  i"  (cot2  A  +  cot2  B  +  cot  A  cot  B) 


CONDITION   OBSERVATIONS. 

Ex.  4.  The  measured  values  of  the  angles  of  a  triangle  have  the  same 
weight.  Show  that  if  the  corrections  to  the  angles  are  expressed  in  terms  of 
the  corrections  to  the  log.  sines  of  the  angles,  and  the  corrections  to  these 
log.  sines  found  by  treating  them  as  observed  quantities,  the  same  results 
will  be  obtained  as  in  Ex.  2. 

For  example  take  50°,  60°,  70"  oo'  30". 

Indirect  Solution — MctJiod  of  Correlates. 

1 10.  If  the  unknowns  in  the  condition  equations  are  much 
entangled  the  direct  solution  would  be  very  laborious.  It 
is  in  general,  therefore,  advisable,  instead  of  eliminating  the 
nc  unknowns  directly,  to  do  so  indirectly  by  means  of  unde- 
termined multipliers,  or  correlates,  as  they  are  called. 

If  we  multiply  the  condition  equations  3,  Art.  108,  in 
order  by  the  correlates  k' ,  k",  .  .  .,  we  may  write 

GO  =  k'  (|>;]  -  /')  -f-  £"(O]  -  O  +  •  •  •  +  O'"]  =  a  min.  (i) 

and  determine  k',  k" ,  .  .  .  accordingly. 
By  differentiation, 

d<*>  =  (a'k1  +  b'k"  -f  .  .  .  -f  2p,vl)dvl 

+  (a"k'  +  b"k"  +  .    .   .  +  2A^Kz'2  +  ...     (2) 

If  we  place  equal  to  zero  the  coefficients  of  nc  of  the  dif- 
ferentials dv^  dvv  .  .  .  we  shall  have  nc  equations  from  which 
to  find  k',  k",  .  .  .  Substitute  these  nc  values  in  the  ex- 
pression lor  </cy,  and  there  will  remain  n  —  ;/,.  differentials 
which  are  independent  of  one  another.  In  order  that  the 
function  may  satisfy  the  condition  of  a  minimum,  the  co- 
efficients of  each  of  these  differentials  must  be  equal  to  zero. 
This  gives  ;/  —  ;/,.  equations,  which  equations,  taken  in  con- 
nection with  the  nc  condition  equations,  give  the  n  unknowns 


The  practical  solution  would,  therefore,  be :  Form  ;/ 
equations  by  placing  equal  to  zero  the  differential  coefficients 
of  the  minimum  function  with  respect  to  each  of  the 
quantities  c\,  7'.,,  .  .  .  TV  From  these  ;/  equations  and  the 
nc  condition  equations  determine  the  n-\-nc  unknowns 
k' ,  k",  .  .  .,  vlf  •:>„,  .  .  .,  and  thence  the  function  [/t/t/]. 


218 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


In  carrying  this  out  the  form  of  the  differential  equation 
2  shows  that  it  would  be  advantageous  to  multiply  the 
minimum  equation  by  —  |,  and  so  write  (i)  in  the  form 


>]  =  a  min.     (3) 
Differentiating,  we  have  the  n  correlate  equations 

a"k'+b"k"+  .  .  .  =/X  (4) 

Substituting  for  z/,,  ZA,,  .  .  .  in  the  condition  equations  their 
values  derived  from  these  equations,  and  the  normal  equa- 
tions result.  Thev  are 


(5) 


.   .  from  (4), 


Solving,  we  obtain  k ',  k" ,  .  .  .,  and  thence  z\, '  v»  . 
and  F,,  Fa,  .  .  .  from  (2),  Art.  108. 

The  normal  equations  may  be  written 

\uaa\k'  -f  \uab~\k"  +  .  .  .  =  I' 


where  ?/„  z^2,  .  .  .  denote  the  reciprocals  of  the  weights 
/„  /2,  ...  The  form  of  these  equations  shows  that  the 
coefficients  \_uaai],  \_uab~],  .  .  .  may  be  computed  as  in  Art. 
85,  the  corresponding  scheme  being 


k' 


CONDITION   OBSERVATIONS.  2 19 

If  the  elimination  of  the  normal  equations  is  performed 
by  the  method  of  substitution  (Art  86),  we  have,  by  col- 
lecting the  first  equations  of  the  successive  groups, 

\tiaa\k'  -f  \iiab~\k"    -f-  \iiac\k'"     -{-...=/' 
+  \iibb.\\k° +  \iibc.i\k'"  +  .  .  .  —r.\ 

+  \ucc.2\k'"  -f  .  .  .  =/'".2  (7) 


where  /',  /".i,  I'". 2,  .  .  .  correspond  to  [#/],  [bl.i],  \cl.2\  .  .  . 
respectively. 

These  equations  being  precisely  similar  in  form  to  Eq.  8, 
Art.  86,  the  elimination  gives  (see  Art.  96) 

I'  -         l          I       l''1      R>    I        l'"'2    R"    I 

~  \uaaY  \ubb.i\          \_ucc. 2\ 

'  ,    Z^e    ,  (8) 

]        r  \_ucc. 2] 


where 


._ 

]  (9) 


/".  I    =  I'R  +  /" 

i"f.2  =  i'R"  +rsn  +  /'"  (10) 


29 


22O  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  i.  Take  that  solved  in  Ex.  i,  Art.  109. 
The  condition  equations  are 

Vi  —  vs  +  v\  =  —  o.  76 

v-t  —  v3  +  7'5  =  —  1.66  (i) 

The  correlate  equations  consequently  are 

k'  =  5^1 

k"  =  iv* 

—  k'  —  k"  =  47/3  (2) 

k'  =  7^4 

k"  =  4v6 

To  form  the  normal  equations  we  may  substitute  for  v\,  Va,  .  .  ,  from  (2) 
in  (i),  or  proceed  by  means  of  the  tabular  form  on  p.  218.     We  find 

0.5929^'  +  o.25/£"      =—0.76 
o.  25/6'      +  0.6429^"  =  —  1.66 

The  solution  of  these  equations  gives 

&'  =  —  0.230        k"  =  —  2.492 
whence,  from  the  correlate  equations, 

vi  =  —  o".os,  v-i  =  —  o".36,  va  =  +  o".6S,  z/4  =  —  o".O3,  V&  =  —  o".62 
Check.  The  results  satisfy  the  condition  equations. 

Ex.  2.  The  angles,  A,  B,  C,  of  a  spherical  triangle  are  measured  with  their 
weights,  /i,  /a,/a  ;  required  their  adjusted  values. 

The  condition  equation  may  be  written  (see  Ex.  2,  Art.  109) 

"<J\  +  v?  +  vs  —  / 

with  t/z'2]  =  a  min- 

The  correlate  equations  are 


and  the  normal  equation 


Hence  the  adjusted  values  are  known. 
When  the  weights  are  equal,  then 


the  same  results  as  in  Ex.  2,  Art.  109. 


CONDITION   OBSERVATIONS.  221 

Note.  If  a  condition  equation  is  of  the  form 


the  weights  of  the  measured  values  being  /,,  /a,  .   .   .  /».  then,  proceeding  as 
in  the  above,  we  have 


This  result  is  very  important  and  will  be  often  referred  to. 

Ex.  3.  At   U.  S.  Coast   Survey  station   Pine  Mt.   the  following  were  the 
angles  observed  between  the  surrounding  stations  in  order  of  azimuth  : 

Jocelyne-Deepwater,  65°  n'  52".  500  weight  3 

Deepwater-Deakyne,  66°  24'  15".  553               "  3 

Deakyne-Burden,  87°  02'  24".  703               "  3 

Burden-Jocelyne,  141"  21'  21".  757               "  i 

required  their  most  probable  values. 

The  condition  to  be  satisfied  is  that  the  sum  of  the  angles  should  be  360'. 
Now, 

sum  of  measured  values  =  359°     59'     54".  513 
theoretical  sum  =  360°     oo'     oo".ooo 

.'.  residual  error  5  ".487 

Hence,  as  in  the  preceding  example, 

i 

correction  to  each  of  first  three  angles  =  -  -  —  3  --  X  5  ".487 

a  +  a  +  s  +  J 


correction  to  fourth  angle  —  X  5  .487 

7?  +  8  +  -3  :  +   l 

-  2".  744 


Ex.  4.  Given  in  a  triangle,  of  which  the  longest  side  is  4  miles,  the  three 
angles  measured  with  equal  care  and  with  the  values  i",  5",  179°  59'  58"  ; 
adjust  the  triangle. 

[Angles  as  small  as  single  seconds  occur  in  practice.  In  the  primary 
triangulation  (1877)  of  the  Great  Lakes,  carried  out  by  the  U.  S.  Engineers,  in 
the  neighborhood  of  the  Chicago  base  two  angles  were  measured  with  values 
o".8is  and  i".i8s  respectively.] 

Ex.  5.  "In  order  to  find  the  content  of  a  piece  of  ground,  I  measured 
with  a  common  circumferentor  and  chain  the  bearings  and  lengths  of  its 
several  sides.  But  upon  casting  up  the  difference  of  latitude  and  departure 


222 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


I  discovered  that  some  error  had  been  contracted  in  taking  the  dimensions. 
Now,  it  is  required  to  compute  the  area  of  this  enclosure  on  the  most  probable 
supposition  of  this  error. 

"  Let   ABCDE   be    a    survey   accurately    protracted    according    to    the 
measured  lengths  and  bearings  of  the  sides  AB,  BC,  .   .   .    A  the  place  of  be- 
ginning, E  of  ending,  A  G  a  meridian,  AF,  FE 
the  errors  in  latitude  and  departure.     Now, 
the  problem  requires  us  to  make  such  changes 
in  the  positions  of  the  points  B,  C,  .   .   .  that 
"    we  may  remove  the  errors  AF,  FE — in  other 
words,    that   E   may   coincide  with   A  ;    and 
these    changes    must    be    made   in    the    most 
We   have,   therefore,    to   fulfil    the   three   following    con- 


Fig.ll 


probable  manner. 
ditions  : 

"All  the  changes  in  departure  must  remove  the  error  in  departure  EF. 

"  All  the  changes  in  latitude  must  remove  the  error  in  latitude  AF. 

"  The  probability  of  these  changes  must  be  a  maximum." 

Let  Oi,  a?,  a3,  .  .  .  ;  3i,  32,  $3,  .  .  .  ;  denote  the  measured  lengths  and 
bearings  of  the  sides  AB,  BC,  .  .  .,  and  xi,  xt,  x3,  .  .  .;y\,yi,y*,  .  .  .  their 
most  probable  corrections. 

Now,  since  the  corrected  latitudes  must  balance,  and  the  corrected  de- 
partures must  also  balance,  we  have  the  conditions 

(ai  +  Xi)  cos  (Si  +  yi)  +  (a?  +  jr2)  cos  (32  +  y-i)  +  .   .   .  =  o 
(rtj  +  JTi)  sin  (3i  +  ;'i)  +  (an  +  *2)  sin  (32  +  y-t)  +  .  .   .  =  o 


or,  reducing  to  the  linear  form, 

cos3i  xi  —  a\  sin3i  y\  +  cos32 
sin3i  Xi  +  ai  cos3j  yi  +  sin32 

with 


—  aa  sin32 
+  aa  cos3a 


.   +  [a  cos  3]  =  o 

.  +  [a  sin  3]  =  o    ' 


[A*2]  +  [gy2]  =  a  minimum, 

the  weights  of  xt,  xz,  .  .  .  ;  y\,  yi,   .  .  .  being  /1(  /2, 
spectively. 

Hence  the  correlate  equations 

,  k'  +  sinSi  k"  —  plxl 

i  k'  +  at  cos3i  k"  —  q\y\ 


qit 


(2) 


and  the  normal  equations 


sin  3  cos  3 


_  faii  sin  ^  cos  3"] ) 


sin  3  cos  31       P*2  sin  3  cos  3 


(  Tsin  3-  cos  31       T 

1L"~7""J~L 


=  —  [a  cos  3] 


=  —  [a  sin  3] 


CONDITION   OBSERVATIONS.  223 

Now  if  we  assume 


as  "  this  seems  best  to  agree  with  the  imperfections  of  the  common  instru- 
ments used  in  surveying,"  the  normal  equations  reduce  to 

K  [a]  —  [a  cos  3] 

k"[a]  =  —  [a  sin  3] 

from  which  k' ,  k"  are  known. 

The  corrections  .TI,  x?,  .  .  .  ;  y\  _r2  .  .  .,  are  known  from  (2). 
The  errors  in  latitude  (see  Eq.  i)  now  reduce  to 

[a  cos  31 

cosS,  xt  —a,  sin3i  r,  =  —a^ — r^ 

[«J 

cos  3] 


cos32  .r2  —  rt.>  sin32  )'•_>  =  —  a2 


w 

and  the  errors  in  departure  to 

sin3,  xi  +  «,  cos3i  n  =  —  «j        r  ,  ~ 

w 

[a  sin  3] 
stn32  jr2  +  <z2  sm32  v2  =  —  a2  — 


Hence  Bowditch's  rule  for  balancing  a  survey:  "  Say  as  the  sum  of  all  the 
distances  is  to  each  particular  distance,  so  is  the  whole  error  in  departure 
to  the  correction  of  the  corresponding  departure,  each  correction  being  so 
applied  as  to  diminish  the  whole  error  in  departure.  Proceed  in  same  way 
for  the  correction  in  latitude." 

This  problem  was  proposed  as  a  prize  question  by  Robert  Patterson,  of 
Philadelphia,  in  vol.  i.  No.  3  of  the  Analyst  or  Mathematical  Museum, 
edited  by  Dr.  Adrain.  of  Reading,  Pa.,  and  published  in  1807.  In  vol.  i. 
No.  4  are  two  solutions — one  by  Bowditch,  to  whom  the  prize  was  awarded, 
and  the  other  by  Dr.  Adrain.  Adrain's  mode  of  solution  is  nearly  the  same 
as  by  the  ordinary  Gaussian  method.  He  employs  undetermined  multipliers 
or  correlates,  exactly  as  Gauss  subsequently  did.  To  Adrain,  therefore,  is 
due  not  only  the  first  derivation  of  the  exponential  law  of  error,  but  its  first 
application  to  geodetic  work.  See  Appendix  I. 


224  THE   ADJUSTMENT   OF   OBSERVATIONS. 

To  Find  the  Precision  of  the  Adjusted  Values,  or  of  any  Func- 

tion of  them. 

in.  The  method  of  proceeding  is  the  same  as  in  Art. 
101. 

The  first  step  is  to  find  //,  the  m.  s.  e.  of  a  single  observa- 
tion, and  next  the  weight,  /v,  of  the  function,  whence  the 
m.  s.  e.  of  the  function  is  given  by 


Up  being  the  reciprocal  of  the  weight. 

(a)  To  find  //. 

In  Art.  99  it  was  shown  that  in  a  system  of  observation 
equations  the  m.  s.  e.  fj.  of  an  observation  of  the  unit  of 
weight  is  found  from 

vv\ 


where  \_pvv\  is  the  sum  of  the  weighted  squares  of  the 
residuals  z>,  n  is  the  number  of  observation  equations,  and 
tif  the  number  of  independent  unknowns. 

Hence  in  a  system  of  condition  equations,  n  being  the 
number  of  observed  quantities  and  nc  the  number  of  con- 
ditions, the  number  of  independent  unknowns  is  n  —  ;/,., 
and 


n—(ri—nc) 


/\pvv\ 

Y  (i) 

Liiroth's  formula  (Art.  99)  may  be  used   as  a  check  on  the 
value  of//. 

Checks  of  \_pvv\. — When  the  number  of  residuals  is  large, 
in  order  to  guard  against  mistakes  \_pvv\  should  be  com- 
puted in  at  least  two  different  ways.  The  following  check 
methods  will  be  found  useful: 


CONDITION   OBSERVATIONS.  225 

(«)  The  correlate  equations  4,  Art.  no,  may  be  written 

IXA  7-,  =:  l/^  a'  k'  +  i/w,  V  k"  +  .   .   . 
l7/a  7'2  =:  Vu,  a"k'  -f  Vn.2  b"k"  +  .   .   . 

Square  and  add,  and 

[/>7'7']  =  \_uaa~\k'  V  -\-  2\uab\k  k"  -f  2\uac\k'k'"    -f-  .  .  . 
+    \ubb~\k"  k"  -f  2J>&r]£"/r'   +  .  .  . 

+    \iicc]k'"k'"  +  ...      (2) 

+  .  .  . 
=  [/'/]  from  (6),  Art.  1  10. 


-  /'y&'  -f  /"yfe" 


R 


1  \ubb.i\ 

I  1    f  „  ,  /,  /,      T  H        1        r , ,  n~\  I         '        ' 


_(/7_          (/".I)'          (/-.2)' 

^ 


[ttbb.l']          \_UCC.2] 

by  addition  attending  to  Eq.  10,  Art.  no. 

This  expression  is  very  readily  computed  from  the  solu- 
tion of  the  correlate  normal  equations,  as  shown  in  Ex.  2 
following.  Compare  the  computation  of  [77']  from  the 
scheme  in  Art.  100. 

The  sum  [/TT]  can  in  general  be  computed  more  rapidly 
by  these  methods  than  by  the  direct  process  of  summing 
the  weighted  squares  of  the  residuals. 

Ex.  i.  The   three  angles  of  a  triangle  are   measured  with   the   weights 
/i,  /a,  /a  /  required  the  mean-square  error  of  a  single  observation. 
Using  the  values  of  r^,  ?•«,  7-3  found  in  Ex.  2,  Art.  no,  we  have 

r.  ......  _"'/2   ,    "*r'   .    "»/a 

-  M*  +  [«]•  +  M. 

-11 

"M 


226 
Hence 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


/ 


//  =  — —  since  nc  =  i 
Check  (i).    [pvv\  =  [/t/] 


as  before. 


, 
~M 


Check  (2).    \_pvv\  =  pr-y  directly  from  Eq.  3,  since  [uaa]  —  i. 
[u] 

Ex.  2.  To  find  the  m.  s.  e.  of  a  single  observation  in  Ex.  i,  Art.  no. 
The  first  step  is  to  find  the  value  of  \_pvv\.     Three  methods  are  given 


p 

1! 

pvv 

5 

—  0.05 

.0125 

7 

—  0.36 

.9072 

4 

+  0.68 

1.8496 

7 

—  0.03 

.0063 

4 

—  0.62 

1.5376 

4.3132  =    [/ZT>] 

(2) 


k 

< 

£/ 

—  O.23O 
-  2.493 

—  0.76 

-  1.66 

0.1748 
4-1384 

4.3132  =  [/?'-'] 

(3)   From  the  solution  of  the  correlate  normal  equations: 


-6' 

k" 

+  0.5929 
+  0.2500 

+  0.2500 
+  0.6429 

—  0.76 
-  1.66 

=     r    } 

r 

+  i. 

+  0.4217 

-  1.2818 

=  —  J 

[u.aai 

+  0.5375 
+  i. 

-  1-3395 

=     /".i     1 

[ul>/>.i\ 

CONDITION   OBSERVATIONS. 
•'•      [/"'''']  =o.j()  X  1.2819  +  1.3395  X  2.492 

=  4.3136 

Hence,  the  number  of  conditions  being  two, 


(b)  To  find  iiF. 

Let  the  function  whose  weight  is  to  be  found  be 

F=f(Vlt  F2,  .   .  .  r«)  (4) 

and  let  it  be  conditioned  by  the  nc  equations 

/,(r,,  rft.  .  .  ru)=0 


Expressing  F  m  terms  of  the  observed  values,  M^M.r  .  .  .  J\f,n 
which  are  independent  of  one  another,  and  reducing  to  the 
linear  form,  we  have 


f)F  f)F 

'+v*+-  •  • 


Hence  as  in  Art.  101, 


where  ?/,,  ?/.,,  .   .  .   are  the  reciprocals  of  the  weights  of  the 
observed  values. 

As  it  usually  requires  a  long  elimination   to  express   F 
in  terms  of  J/,,  M»,  .   .  .  MM  direct!}7,  it  is  better  to  compute 

:,  .  .  .  trom  the  torms 


6M 


HL         *_F    i^_.     ()F    :)F'    i 
o  J/,  ~"  o  I  \  oM,  "•"  o  F2  r)  J/,  ~" 

rtF        oF    f)  F 


228  THE   ADJUSTMENT    OF    OBSERVATIONS. 

Ex.  3.  To  find  the  m.  s.  e.  of  a  side,  a,  in  a  triangle  wliose  angles  have 
been  measured  with  the  weights  />,,  f^,  f>3,  the  base,  />,  being  free  from  error. 
The  function  equation  is 

sin  A 

F=a  =  b  - — 
sin  B 

and  the  condition  equation 

A  +  B  +  C=  180  +  £ 
Hence  from  Ex.  2,  Art.  no,  expressing  A,  B  in  terms  of  the  observed  values, 


B  =  M,  +  j-? j  180  +  £  -  (Aft  +  M,  +  M,)  • 
Now, 

=  a  sin  i"  /  •<  (  i  —  r— n  )  cot  A  +  ~.  cot  B\-v\ 
\\\         [«]/  [«]  i 

I  «1  /  «S\  )  I  #1  W-j 

+   ^    -  j^j  COt  ^   -    (  I  -  -        }    COt  B  -7'2  +    -]    -  f—   COt  /i     +j— : 
I          M  V  M/  i  <          M  L«] 

Therefore 

„  /  \  /          «i  \  «••  I  ~ 

IIF  =  a'J  sin-  i    I  -  I  i  —  p—  1  cot  A  +  j— -=  cot  j9  v  ?/, 


=  rt2  sin-  i"  -,  (  «,  —     "-:  |  cot'2^  +  (  «a  —  r^r  ) 

(  V       L«]  /  \       H  / 


2  cot  ^  cot 


and  jiif  =  jii  \'up 

where  //  is  the  m.  s.  e.  of  a  single  observation. 

If  the  weights/i,  p*,  />3  are  each  equal  to  unit\r,  this  reduces  to 

tip'1  =  |  a"  sin2  1"  /<2  (cot2^  +  cot2.5  +  cot  A  cot  B) 
and  if  the  triangle  is  equilateral, 

[<F-  =  3  a*  sin2  1"  //2 

Also,  if  the  base,  instead  of  being  considered   exact,  had  the  m.  s.  e.  ///,, 

«2 
the  expressions  for  /ip-  would   be  increased   by     „  in"  and  //*'-  respectively. 


CONDITION   OBSERVATIONS.  229 

It  is,  however,  usually  much  more  convenient  in  practice 
to  use  the  method  of  correlates. 

Let  the  function,  reduced  to  the  linear  form,  be  written 

dfe/X +/'»,+  .   .   .  (9) 

This  is  conditioned   by   the  nc  equations,  also  in  the  linear 
form, 

a'i\  -f-  a""i'-.  ~\-  •   •   •  —  /'  =  o 

/A'1-{-£V2+  .   .   .  -/"=o  (10) 

with  » 

\ pi>i'\  =  a  minimum. 

Referring  to  the  principle  of  Art.  1 10,  we  see  that  by 
using  correlates  /£•',  k" ,  .  .  .,  and  determining  them  properly, 
we  can  express  the  function  in  terms  of  the  quantities 
?',,  i>.2,  .  .  .  vn  as  it  independent;  that  is, 

dF=  (/'  -  a'k'  -  b'k"  -  .   .   .)  r, 

+  (/"  -a'k'-b'k*-  .    .    >,+  .    .    .      (II) 

and,  therefore, 

HF=  (f  —  a'k'  —  b'k"  --  .   .   .)'"'//, 

+  (/"  -a!'k'-b"k"  •-  .   .   .)X+-   •   •     (12) 

It  remains  to  determine  k' ',  k' ',  .  .  .  Now,  when  the  most 
probable  values  of  the  corrections  z/,,  v^  .  .  .  i'n  are  sub- 
stituted in  the  value  of  the  function  dF,  this  function  must 
have  its  most  probable  value,  and,  therefore,  its  maximum 
weight.  We  may,  therefore,  determine  the  correlates  k 
from  the  condition  that  the  weight  of  dF  is  a  maximum  ; 
that  is,  that  UF  is  a  minimum.  Differentiate,  then,  UF  with 
respect  to  k' ,  k",  ...  as  independent  variables,  and  we  have 
the  equations 

\uaa\k'  -f-  \iiab\k"  -f-  .   .   .  =  \naf~\ 
[uaW+[uMW+  •   •   •  =\«&rt  (13) 

from  which  k ',  k" ,  .   .   .  are  found. 


230 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


These  equations  being  precisely  of  the  form  of  ordinary 
normal  equations,  it  follows,  as  in  (c)  and  (d),  Art.  100, 
that 

»F  =  Wf\-\.™f\k'  -  \ubf\k"  -  (14) 

or 

it    —  r  H  ff~\  (  T  c*  \ 

tip  —     **JJ    \     -  -p  -,  .  7      -.  ...  v  !  b  / 

[uaa]        \ubb.  i  J 

The  form  of  the  last  expression  for  up  shows  that  it  ma)' 
be  found  by  means  of  the  following  scheme,  in  which 
[«#/],  \_ubf~],  .  .  .  are  added  as  an  extra  column  in  the 
solution  of  the  correlate  normal  equations  (13),  in  the  man- 
ner shown  in  Art.  100.  For  three  correlates  the  scheme 
would  be 


V 

k" 

k'"- 

\}iad\ 

'  \jiaU\ 

\jiac\ 

\_naf\ 

\iibU\ 

\_ubc] 

w\ 

[we] 

\_rnf  -\ 

\ubb,\\ 

\ubc.i] 

\_ucc.i] 

\UCf.l\ 

< 

[HCC.2] 

w^ 

=  Up 

CONDITION    OBSERVATIONS. 


231 


Ex.  4.  To  find  the  weight  of  the  angle  PSB  in  Ex.  i,  Art.  109. 
Here 

dF  =  —vi  +  7'3 
.-./,=  -!,      f-2  =o,      /3-  +  i 

From  the  condition  equations 


a  =  +  i 
a"''  =  —  i 
a""  =  +  I 


b"  =  +  i 
//"  =  -  i 
//""  =  +  i 


.-.     [uaf]  =  J  X  -  i  +  i  X 
[«*/]=-* 
[«</]  =  +  0.45 


-  0.45 


The  correlate  normal  equations  with  the  extra  column  for  finding  UF: 


k' 

k" 

' 

+  0.5929 
+  0.2500 

+  I 

+  0.2500 
4-  0.6429 

+  0.4217 
+  0.5375 

—  o.  7600 
—  I.  6600 

—  1.2818 

-  1-3395 

—  0.4500  —  [>?/l 
+  0.2500  =  [itl>f~\ 

-0.7590 
—  c.o6o2  —  [ul>/.i] 

+  0.4500  =  [«//] 
+  0.3416 

+  0.1084  =[«//•!] 

+  I 

—  2.492 

—  0.  1  1  2O 

4-  0.0067 

+  O.IOI7  =[tlff.2\ 

Also 


as  before. 


=  1.47  \  0.1017  from  Ex. 
=  o".47 


Ex.  5.  To  find  the  %veight  and  m.  s.  e.  of  the  adjusted  value  of  an  angle 
of  a  triangle  when  all  three  angles  are  measured,  their  weights  being/,, /2,/3 
respectively. 

The  function  is 

dF  =  r, 

and  the  condition  equation 


232  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Hence  from  (15) 


M 

Also  /<-=  /'  Vu 


/  / — I \ 

~~^\Vr-\uT     <SeeEx-1- 


The  weight  of  an  angle  before  adjustment  is  to  the  weight  after  adjustment,  as 

1.       M 

Ml    '     #l(«2  +   U3) 


If^i,  =/„  =/>3  =  i,  the  weights  are  as  2  :  3.  This  result  is  independent  of 
the  magnitude  of  the  angle.  It  therefore  applies  to  any  problem  in  which 
the  condition  to  be  satisfied  is  that  the  sum  of  two  quantities  shall  be  equal  to 
a  third,  or  in  which  the  sum  of  all  three  is  equal  to  a  constant.  For  other 
solutions  see  Ex.  2,  Art.  109. 

Ex.  6.  If  n  angles  measured  at  a  station  close  the  horizon,  find  the  weight 
of  the  adjusted  value  of  any  one  of  them. 

[The  solution  is  exactly  as  in  the  preceding  example. 
The  weight  of  Vlt  for  instance,  is  found  from 


1  1  the  weights  /i,  /a>  .  .   .  are   all   equal  to  one  another,  the  weight  of  an  angle 
after  adjustment  is  to  its  weight  before  adjustment  as 

n  :  n  —  i      ] 

Ex.  j.   Show  that  the  weight  of  the  sum  of  the  adjusted  angles  of  a  triangle 
is  infinite. 

[Sum  =  180  +  e  ,  a  fixed  quantity, 

.  ".     m.  s.  e.  =o,  and  weight  =  co 
or  otherwise 


CONDITION   OBSERVATIONS.  233 

Ex.  8.  In  the  "longitude  triangle"  Brest,  Greenwich,  Paris,  as  de- 
termined by  the  U.  S.  Coast  Survey  in  1872,  the  observed  values  were 

m.      s. 

Brest-Greenwich,  17  57.154  weight  10 
Greenwich-Paris,  9  21.120  "  7 
Brest-Paris,  27  18.190  "  9 

Show  that  the  most  probable  values  are 

m.     s. 

T7  S7-13°     weight  14 

9  21.086          "        12 

27  18.216          "        13 

Ex.  9.  To  find  the  weight  of  a  side  in  a  chain  of  triangles,  all  of  the  angles 
of  each  triangle  having  been  equally  well  measured  and  the  base  being  free 
from  error. 

Let  A  be  the  measured  value  of  the  base,  and  let  «- j- ^ 

Hi,  a-i,   .   .   .  <rn  be  the  sides  of  continuation  in  order  as  \     /  \    /    \ft 

computed  from  b  ;  an  being  the  side  whose  weight   is      /A,    B\/       \/ 
required.  Fig. 12 

If  A:,  Bi,  A^,  B-i,  .  .   .  are  the  measured   values 

of  the  angles  used  in   computing  an  from  />,  the  angles  A,,  A-2,  .  .   .   being 
opposite  to  the  sides  of  continuation,  then 

Hi  _  sin  A  i    a-i  _  sin  AI  an      _  sin  An 

/>        sin  Bi    c/i       sin  B?    '         '  nn  -  \        sin  fin 

Hence  by  multiplying  these  expressions  together, 

_     sin  Ai   sin  Ai  sin  An 

sin  Bi  sin  £-2  sin  Bn  (r) 

We  may  now  proceed  in  two  ways. 
(a)   Differentiating  directly, 

da*  —  an  sin  i"  [cot  A  (A)  —  cot  B  (B)] 
where  (A),  (/?),  .  .  .  denote  the  corrections  to  A,  B,   .   .  . 

[In  a  chain   of  triangles  it  is  convenient  to  use  the  notation  (A),  (B)  .   .   . 

for  T'i,  v-i the  parentheses  indicating  corrections.] 

The  condition  equations,  from  the  closure  of  the  triangles,  are 

/    f    \     i     /  /?   A     i     /  /"*  \  ' ' 

v'  i  /  T  1-^-2  i )  ~r  \L*  i )  —  i 

Substituting  in  Eq,  15, 

tian  =  I  <?«-  sin-  i"  [cot-  A  +  cot-  B  +  cot  A  cot  B\  (3) 

the  result  required. 

If  the  triangles  are  equilateral  this  reduces  to 

Ma>i  =  *  n/>-  sin-  i"  (4) 


234  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Hence  in  a  chain  of  equilateral  triangles  the  weights  of  the  sides  decrease  as 
we  proceed  from  the  base,  />,  through  the  successive  triangles,  inversely  as  the 
number  of  triangles  passed  over  ;  that  is,  are  as  the  fractions 

1,  i,  i  i,   •  •   • 
(b)  Taking  logs,  of  both  members  of  Eq.  I  and  differentiating, 

«/log  an  —  -^—  log  sin  Ai  (Ai)  —  — -  log  sin  J5t  (Bt)  +  .   .   . 
a  A  i  (Mi  i 

=  [8A(A)-8B(B)\  (5) 

or  expanding  the  first  member, 

dandaH=\8A(A)-8B(B)~\  (6) 

where  Sa  is  the  tabular  difference  for  one  unit  for  the  number  an,  and  8 A,  SB 
are  the  logarithmic  differences  corresponding  to  i"  for  the  angles  A,  B  in  a 
table  of  log.  sines.  (See  Art.  7.) 

Hence  attending  to  the  condition  equations  2,  we  have  from  (15) 
for  Eq.  5, 

Ulogan  =  It^V  +  SA  $B  +  S*~] 

and  for  Eq.  6, 

/r  +  8 A  SB  +  SB"] 


as  giving  the  weight  of  the  logarithm   of  the  side  and   the  weight   of  the   side 
respectively. 

Of  the  two  forms  (a)  and  (b),  the  logarithmic  is  in  general  the  most  con- 
venient in  practice. 

Ex.  10.  From  a  baseAB(=/>)  proceeds  a  chain   of  equilateral  triangles, 
B  all    of    the   angles    being    equally 

~7\t^<     BA  /       well  measured,  and  the  sides  BC, 

•>    /    \    /      \    /      \    /        C^'  •  •  •  being  in  the  same  straight 

VA» BA/ V  line.     Find  the  in.  s.  e.  of  the  line 

BN,  which  is  n  times  the  base. 
Take  first  the  simple  case  of  n  =  2. 

sin   C\          sin  Ai  sin  A?  sin  C* 

F  =  BN  =b  — h  l>  — 

sin  Bi          sin  Bi  sin  -B-i  sin  B3 

—  cot £3(83)  +  cot  C3  (C3)\  6  sin  i" 
Also,  we  have  the  condition  equations 


CONDITION   OBSERVATIONS.  235 

Hence 

1<H  =  3       [«/J    =o 
[W.i]  =  3       0/.i]=o 

O.2]  =  3         [</.2]=0 

[//J    =  (cot-  .-/  1  +  4  cot-  j9,  4-  cot2  C,  +  cot'-  .-/.j  4-  cot-  £,  +  cot-  /^  4-  cot-  G)/'-  sin*  i  " 
—  '.j0  /''-  sin'-  i"       since  cot'-'  Go"  =  .\- 

Substituting  in  Eq.  15, 
and  therefore 


where  /i  is  the  m.  s.  e.  of  an  observed  angle. 
Generally, 

dF=(n  —  i)  cot,-/,  (A,)  —  n  cot  A  (#,)  +  cot  C,  (C,) 
4-  (w  -  i)  cot  A-,  (//,.)  —  («  —  i)  cot  j92  (B.S) 

—  («  —  i)  cot  B*  (/>',)  4-  cot  Cs  (C3) 
+         .......... 

and 


If  the   cliain    proceeds  in   the   opposite  direction   until  At\"=BA',   then 
since  n  .(iVJ  =//A,V'-',  and  ^V.\T'  =  2/>«  approximately,  we  have 


/  A        sn  i4 
If  NN'  is  «  times  the  base  |  putting  ;/  =  -  j 


,.    /2/r  —  yt 
jUtfX>  =n  AN  sin  i  4/  - 


+•  io 
iSw 


Hence  it  follows  th.it  in  a  chain  of  ccjuilateral  triangles  whete  one  base  only 
is  measured,  it  is  better  to  place  the  base  at  the  centre  of  the  chain  rather  than 
at  either  end. 

£.r.  ii.  If  a  chain  of  equilateral  triangles  proceeds  from  the  base  AB,  as 
the  chain  in  Ex.  io,  but  in  the  opposite  direction,  show  that  the  m.  s.  e.  of 
BN'  ,  which  is  ;/  times  the  base,  is 


and  if  the  chain  proceeds  also  in  the  opposite  direction  until  AX—-BX,  then 
if  XX'  be  taken  ;;  times  the  base, 

//VA.,  =  //  AW  sin  i"4  /2//'' 
31 


236  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  12.   If  a  chain   of  equilateral  triangles   proceeds  from  the  base  AB, 

which  is  in  the  same  straight  line  as 

r£ A B N      the   derived   side  BN,    show  that  the 

in.  s.  e.  of  AN,  which  is  (n  +  i)  times 

the  base  b,  is 

V  V 

Fig.14 


—  sin  i "  V4«:i  +  gn-  +  5« 


tsin^]  sin  Ai  sin  C3          sinAi  sinA?  sin^43  sinA*  sin  C5  "1 

F  =  b  +  b  —  —  h  b  —  -  + 

sin£i  sin/y2  sinB3          sin  BI  s\nB-2  sin  B3  sin/?4  sinj95  J 


If  also  a  chain  of  equilateral  triangles  proceeds  from  AB  in  the  opposite 
direction  to  JV',  then  if  NN'  is  «  times    the  base,  show  that  the  m.  s.  e.   of 

NN'  is 


f.i  NN 


'  sin  I"A/[ 


n1  +  3«  —  4 
gn 


Hence  show  that  in  computing  a  line  NN'  ,  equal  to  n  times  the  base  AB, 
through  a  chain  of  equilateral  triangles,  the  least  loss  of  precision  is  with  the 
form  of  Fig.  12. 

Ex.  13.  To  find  the  m.  s.  e.  of  the  altitude  of  a  triangle,  the  base,  l>,  being 
supposed  free  from  error,  and  the  reciprocal  weights  of  the  angles  being 
7/1,  7/2,  ua  respectively. 

The  function  is 

,  sin^4  sinC 
F=b  -  :  —  — 
sin  j? 

.'.  dF=Fs\n  i"|cot^  (A)  —  co\.B  (B)  +  colC(C)\ 

Also  the  condition 

(A)  +  (B)  +  (C)  =  I 
Substituting  in  (15) 

UF  =  F^  \  ?/i  cot5  A  +  7/2  cot'2  B  +  7/s  cot2  C 

__  (T/I  cot  A  —  7/2  cot  B  +  7/3  cot  C)2 


If  Z  A  =  Z  C,  and  u^  =  7/2  =  7/3=- 

/ 
then 

2   F- 
HF--    —    sin-  i    cosec2./? 

and 

Hb  sin  i"  B 

/IF-      .,—  -  cosec2- 


where//  is  the  m.  s.  e.  of  the  angle  corresponding  to  the  unit  of  weight. 


CONDITION    OBSERVATIONS. 


237 


E.\.  14.   If  two  similar  isosceles  triangles  on   opposite  sides  of  the  base 
ACarc  measured   independently,  thus  forming  a  rhombus 
(vertices  B,B'),  then,  taking  the  weight  of  each  angle  unity, 

f.tb  sin  i"  0  B 

/<««•=   f^   ~  —  cosec-  — 


r> 

and  if  KB'  is  n  times  the  base  /',  then,  since  cot  —  =  n, 

2 


_  sin  i 

2  V3 

Caution. — If    we    solved    for   the    rhombus    directly    it 
would  not  do  to  take 

D 

BB'  =  b  cot  - 

2 

and  then  form  HBB>.  The  result  would  be  V  2  times  too  great.  For  as  the 
triangles  are  measured  independently,  each  half  of  BB'  must  be  considered 
separately,  so  that  we  must  use  the  form 


BB'  =  -  (  cot  —  +  cot  —  ) 

2  \  2  2    / 


with  the  condition  equations 

(A)  +  (B)  +  (C)    =  /, 
(A1)  +  (B')  +  (C)  =  /, 

corresponding  to  the  angles  of  the  two  triangles. 

Ex.  15.  If  on  a  b.ise,  b,  as  diagonal  two  similar  isosceles  triangles  are 
described,  forming  a  rhombus,  and  on  the  other  diagonal  of  this  rhombus  two 
triangles  similar  to  the  former  are  described,  forming  a  second  rhombus,  and 
so  on  /«  times,  required  th'j  m.  s.  e.  of  the  last  diagonal,  all  of  the  angles 
being  equally  well  measured. 

For  the  »itk  diagonal  </ 


cot  —   4-  c 


y?2W 
ot  —    I 

2  /   \ 


cot   —  h  cot 


Fig.16 


are    the    vertical    angles    in 


where    B\,   />'...,   . 
order. 

Now,  as  the  triangles  are  all  similar, 

/,',  =  />.,  =r  />':,  —  .   .   .  =  A'.j,,,  =  B  suppose. 
Hence 


b  B  Ji   .       , 

f    •=.  f    =...=  -  cot"1"1  —  cosec-  — sin  i 
422 


238  THE    ADJUSTMENT    OF   OBSERVATIONS. 

and 

(  {/>"  /> 

tid  =  2m-\  I  —  cot2'"-2— 

j  Vi6  2 


inb~-  />'  .  II    .    .,     „ 

--    cot2'""2  —  cosec  —  sin-  i 

But 

d—b  cot'"  — 


If  d  is  »  times  the  base, 


For  further  development   ol   this  subject  consult   Helmert,   Stiuiicn    fiber 
rationelle  Vennessnngtn.      Leipzig,  1868. 


Solution  in   Two  Groups. 

112.  In  geodetic  work  it  often  happens  that  the  observed 
quantities  are  subject  to  a  simple  set  of  conditions  which 
may  be  readily  solved  as  observation  equations  by  the 
method  of  independent  unknowns,  and  are  also  subject  to 
other  conditions  which  are  best  solved  by  the  method  of 
correlates.  The  equations  are  thus  divided  into  two  groups 
for  solution,  and  the  complete  solution,  therefore,  consists 
of  two  parts.  The  observation  equations  forming  the  first 
group  are  solved  by  themselves  and  give  approximations  to 
the  final  values  of  the  unknowns.  The  corrections  to  these 
approximate  values  due  to  the  second  group  are  next  found 
by  solving  this  second  group  by  the  method  of  correlates.* 

The  merit  of  the  method  consists  in  utilizing  the  work 
expended  in  the  solution  of  the  first  group  in  determining 
the  additional  corrections  due  to  the  second  group.  The 

*  The  first  exposition  of  this  method  was  given  by  ]>esscl  in  the  Gradmcssiingin  Ostfrcusscn. 
The  method  of  finding  the  precision  of  the  adjusted  values  is  due  to  Andrae,  Den  Danskc  Graii- 
Htaaling,  vol.  i.  Very  complete  statements  will  lie  found  in  the  introduction  to  Die  frcussischf 
T.andestriangiiliitiitii)  vol.  i.,  I!erlin,  1874;  Ferrero,  Exposizionedcl  metoda  dci  minimi  quaii- 
rati,  Florence,  1876;  Jordan,  Hamibuch  der  I'crmcssungskundc,  Stuttgart,  1878. 


CONDITION  OBSERVATIONS.  239 

solution  is  rigorous,  and,  being  broken  into  two  purls,  is 
more  eusily  managed  than  if  all  of"  the  equations  had  been 
solved  simultaneously. 

Let  the  first  group  of  equations  be  ihe  observation  equa- 
tions, u  in  number  and  containing  ;/„  unknowns  (;/  >  //„), 

",-i'  +  b,y  -f  .  .  .  -  /,  =  c\     weight  /, 

/+.  •  •  -l*  —  v,        "     p.,          ID 


and  the  second  group  the  condition  equations,  nc  in  number, 
involving  the  same  unknowns  (uc  <  //„), 

a  x  -j-  a"y  -]-...  —  /'  =  o 

b'x-{-b"y-\-  .   .   .  —  I"  —  -  o  (3) 

The    most    probable   values   of    the    unknowns  .r,  j',  .    .   .   are 
those  which  are  given  by  the  relation 

[/TV]  =  a  minimum.  (^3) 

It  is  required  to  find  them. 

The  value  of  an  unknown  is  found  in  two  parts,  the  first, 
(.r),  (j'),  .  .  .,  arising  from  the  observation  equations,  and 
the  second,  (i),  (2),  .  .  .,  arising  from  the  condition  equa- 
tions, thus  : 


Now,  overlooking  for  the  present  the  condition  equations 
and  taking  the  observation  equations  only,  (.r),  (j-i,  •  .  . 
would  be  found  by  solving  these  equations  in  the  usual 
wav.  \Ve  have,  therefore,  reducing  all  to  weight  unitv  for 
convenience  in  writing,  the  normal  equations 


«l>  |C.r)  -f  [bb  ](;')+•  •   --l/'/i  (5) 


240  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  solution  of  these  equations  gives  (see  Art.  97) 


Hence  (x),  (j'),  .  .   .  are  known. 

To  find  the  condition  corrections  (i),  (2),  .  .  .  ,  eliminate 
T'J,  7'.,,  .  .  .  i'n  by  substituting"  in  the  minimum  equation, 
which  then  becomes 

\ad\xx-\-2\ab~\xy-\-  .  .  .  —  2\_al~\x 

+    Ww+.  -  -   -2[£/>  (7) 

+  [//]  =  a  rain. 

This   equation    is   conditioned   by    equations  2.      Thus    the 
solution  is  reduced  to  that  already  carried  out  in  Art.  no. 

Calling  /,  //,...  the  correlates  of  equations  2,  we 
have  the  correlate  equations 

|  \ni\x  -f-  \aU\  y  +  .  .  .  —[al~\  =  a'I+ 
]x  +  \bb\y  +.   .   .  -[l>r\  =  a"I+ 


These  equations,  taken  with  (4)  and  (5),  give  the  relations 


aa 


suppose 

2  i        »      (8) 

which  being  of  the  same  form  as  (5),  their  solution  gives 

or  substituting  for  |  i  |,  \2\,  .    .  .   their  values  from  (8), 

(i)  — A'/+B'//-|-C'///  +  .    .    . 

CV//+  ...  (10) 


CONDITION    OBSERVATIONS.  241 

where 

A'  =  \aa~\a'  -\-  \a,i\a"  -)-  .   .   . 

15  '  =  \tuL\b'  -f  [>V  |//'  +  .  .  •  (\  i  ) 

and  arc  known  quantities. 

We  have,  therefore,  expressed  the  corrections  (i),  (2),  .  .  . 
in  terms  of  the  unknown  correlates  /,  //,...  It  remains 
now  to  find  these  correlates. 

Substituting  for  x,  y,  .  .  .  their  values  from  (4)  in  the 
condition  equations,  and 

«'(!)+„"  (2)+.    .    .=/.' 

//(!)  +  £"  (2)+.     .    .    =//  (12) 

where 


and  are,  therefore,  known  quantities,  since  (x],  (j')>  •  •  •  are 
known. 

Substitute  the  values   of  (i),  (2),  .   .  .,  from  (10)  in  (12), 
and  we  have  the  correlate  normal  equations 

[rtA]  I  -\-\_a~\i\  //+...  =/.' 
^T//4-.   •   •  ~- 


where 

[rt  A]  = 


etc.  =  etc. 


(15) 


The  solution  of  these  equations  gives  the  correlates 
/,  //,  .  .  .  Hence  the  corrections  (i),  (2),  .  .  .  are  known. 
Also,  since  (V),  (j),  .  .  .  have  been  found  from  (6),  the  total 
corrections  ,r,  j,  .  .  .  are  known. 


242  THE    ADJUSTMENT    OF    OBSERVATIONS. 

113.    In    carrying   the  preceding  solution    into    practice 
the  following  order  of  procedure  will  be  found  convenient: 

(a)  The  formation  and  solution  of  the  observation  equa- 
tions (i). 

The  partially  adjusted   resulting  values  (V),  (y\  .   .  .  are 
now  to  be  used. 

(b)  The  formation  of  the  condition  equations  (12). 

*''(!)  +  „' (2)+.    .    .   =/„' 

//(!)  +  // (2)  +  .  .  .=/: 

(c)  The   formation  of  the  weight  equations   (9).     They 
are  at  once  written  down  irom   the  general  solution    of  the 
observation  equations  in  (a),  and  are 


(d)  The  formation  of  the  correlate  equations  (8). 

|Tj  =  *'/+£'//+  .  .   . 
IT  —  a"I+b"  11+  .  .  . 

(e)  The  expression   of  the  corrections  in   terms  of  the 
correlates  by  substituting  from  (d)  in  (c). 


(2)  =  AV+B*//+.    .    . 

(f  )  The  formation  of  the  normal  equations  by  substitut- 
ing from  (e)  in  (b).     They  are, 


(g)  The  determination  of  the  corrections  by  substituting 
the  values  of  the  correlates  in  (e). 


CONDITION   OBSERVATIONS.  243 

1  14.  To    Find    the    Precision    of   the    Adjusted 
Values  or  of  any  Function  of  them. 

(a)  First  find  //,  the  m.  s.  e.  of  an  observation  of  weight 
unity. 

We  have  (Art.  m) 


number  ot  conditions 


since  ;/  —  ;/„  is  the  number  of  conditions  in  the  observation 
equations,  and  //,.  the  number  in  the  condition  equations. 
To  find  [vv~\.     From  the  first  observation  equation 


Similarl 


where 


that  is,  T',",  z'2°,  .   .   .   are   the   residuals  arising  from  taking 
the  observation  equations  only. 

Attending  to  Eq.  5,  p.  239,  it  follows  evidently  that 

[*7'°]  =  O        [^'°]  =  O,    .     .     . 

Square  the  residuals  7',,  TV,,  .   .   .  and  add,  then 


=  [''"''"]  +  [wzt'J  suppose. 

The  total  sum  [7-7-]  may  therefore  be  found  in  two  parts,  one 
from  squaring  the  residuals  of  the  observation  equations, 
and  the  other  from  the  corrections  (i),  (2),  .  .  . 


244  THE   ADJUSTMENT   OF   OBSERVATIONS. 

We  proceed  to  put  \ww\  in  a  more  convenient  shape  for 
computation. 


from  Eq.  8,  p.  240. 

Substitute  for  (i),  |  1  1  ,  (2),  .  .  .  their  values   from   equa- 

tions S  and  10,  and  expand  ;  then 


which  may  be  transformed,  by  means  of  Eq.    14,  into  the 
form 

[aw]  =  /„'/+/."//+.   .   . 

or,  as  in  Art.  m,  into  the  form 

(/„')*  (//.I)'         (^.2)' 

[aw]  =  p^  +  ==£,  +  r  --  4  +  •  •  • 


These  forms  may  be  readily  computed  as  in  Art.  100. 

(b)  Next  find  the  weight  of  the  given  function  of  the 
adjusted  values. 

Let  the  function,  reduced  to  the  linear  form,  be 

dF=gs+g^+.  .  .  (16) 

where  «£•,,£•„,  .   .   .  are  known  quantities. 

Put  for  x,  }>,  .   .   .  their  values  (V)-|-(i),  (j')  +  (2)»  •   •   • 
and 


Put  for  (i),  (2),  .   .   .   their  values  from  (10),  and 

...      (17) 


CONDITION   OBSERVATIONS.  245 

where  /,  //,...  are  found  from  the  equations 


Using   the    multipliers   £j,   4,  ...    in   order   to   eliminate 
/,  //,  .  .  .,  we  have,  as  in  Art.  in, 

.   .   .  +/„'*,  +  O&.+  .   .   . 


(18) 

We  may  determine  £,,  £2,  ...  so  as  to  cause  the  co- 
efficients of/,  //,...  to  vanish  ;  that  is,  so  as  to  satisfy  the 
equations 

[diX]^  -f  DzB^yf  .   .   .  =  [£A] 


and  then  we  shall  have 

JF  =&(*)+&( 

Substitute  for  /„',  /u",  .   .   .   from  (13),  and 


G&)  +  ^(7)  +  •    •    •  (20) 

where 

Gl=gl—  a'kl  —b'k^  —  .   .  . 
G^g^-a"k,-b"k,-.  .  .  (21) 

We  have  thus  expressed  the  function  in  terms  of  (-r),  (f),  .  .  . 
and  known  quantities. 

Now,  since  (,r),  (y),  .  .  .   are  not  independent,  but  are 
connected  by  the  equations 


the  problem  is  reduced  to  that  i^ready  solved  in  Art.  101. 


246  THE   ADJUSTMENT   OF   OBSERVATIONS. 

If,  therefore,  nF  is  the   reciprocal  of  the  required  weight, 

«*=  [GQ]  (22) 

where 

Ql=[aa-]Gl  +  [ap]Gt+.    .   . 

.  -  .          (23) 


the  quantities  [««],  [«/3],  •  •   •  being  as  in  the  weight  equa- 
tions 9. 

Putting  for   Glt  G9,  .  .  .   their  values  from  (21)  in  these 
equations,  and  attending  to  (11),  we  find 

Si^i-A'^-B'/fc,-  .    .    . 

0,  =  ft-A^1-B^;-.  .  .  (24) 

where 

?i  =  [««Ui  +  [«£]&,+  •  •  • 

••  •  •  (25) 


Substituting  in  (22)  for  C,,  G"9,  .  .  .  <2n  (2a»  •  •   •  their  values 
from  (21)  and  (24), 


But  from  (11)  and  (25) 


Hence,  attending  to  (19),  the  above  expression  reduces  to 

\GG\ 


or  to 


\cc.2\ 


CONDITION    OBSERVATIONS.  247 

To  compute  [^7].     Multiply  each  of  equations  25,  in  order, 
by  g»  gv  •  •  •»  and  add,  and 


where  [««],  [«/5J,  .  .   .  may  be  taken  from  the  weight  equa- 
tions. 

The  remaining  terms  of  the  second  form  of  \GQ\  may  be 
found  from  the  solution  of  the  normal  equations,  as  shown 
in  Art.  in. 


Solution  by  Successive  Approximation. 

115.  This  method  of  solution  (due  to  Gauss)  is  of  the 
greatest  importance  in  adjustments  involving  many  con- 
ditions. It  may  be  stated  as  follows  : 

The  condition  equations  may  be  divided  into  groups, 
and  the  groups  solved  in  any  order  we  please.  Each  suc- 
cessive group  will  give  corrections  to  the  values  furnished 
by  the  preceding  groups,  and  the  corrected  values  will  be 
closer  and  closer  approximations  to  the  most  probable 
values  which  would  be  found  from  the  simultaneous  solu- 
tion of  all  the  groups. 

For  suppose  we  have  the  condition  equations 

a'v.  +  d'v^  .   .   .  =  4 

yVl+yv,  +  .  .  .=ib 

h'v,+h"vt+.   .  .=/» 
k'v,  +  k"v,  +  .   .   .  —  4 

with 

v*   =a  min. 


248  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Let  z//,  v,',  ...  be  the  values  of  vlt  v,,  .  .  .   obtained  from 
solving  the  first  group  alone  ;  that  is,  from 


=  a  mn. 


If  now  (?>/),  (z//),  .   .   .  are   the  corrections  to  these  values 
resulting  from  the  remaining  equations,  then  since 


the  condition  equations  are  reduced  to 


with 


and  the  values  of  (z/)  found  from  the  simultaneous  solution 
of  these  equations,  added  to  the  values  of  z/  found  from  the 
solution  of  the  first  set,  would  be  equal  to  the  value  of  v 
found  directly. 

Similarly  if  v"  ,  v"  ,  ...  be  the  values  of  (v'}  obtained  by 
solving  the  second  set  alone,  and  (v"),  (?'/)»  •  •  •  be  the  cor- 
rections to  these  values  resulting  from  the  remaining  equa- 
tions, then  since 

O2]  =  O'2]  +  O-]  +  [X*")1] 

the  condition  equations  are  reduced  to 
«'«)  +  « 


CONDITION   OBSERVATIONS.  249 

with 

[/(>")']  =  a  min. 

The  quantities  [/£>"],  \_pvnv\,  •  •  •  being  positive,  the  mini- 
mum equation  is  reduced  with  the  solution  of  each  set,  and 
thus  we  gradually  approach  the  most  probable  set  of  values. 
Beginning  with  the  first  set  a  second  time,  and  solving 
through  again,  we  should  reduce  the  minimum  equation 
still  farther,  and  by  continuing  the  process  we  shall  finally 
reach  the  same  result  as  that  obtained  from  the  rigorous 
solution.  In  practice  the  first  approximation  is  in  general 
close  enough. 

It  is  plain  that  the  most  probable  values  can  be  found 
after  any  approximation  by  solving  simultaneously  the 
whole  of  the  groups,  using  the  values  already  found  as 
approximations  to  these  most  probable  values. 

Examples  will  be  found  in  the  next  chapter. 


CHAPTER  VI. 

APPLICATION   TO   THE   ADJUSTMENT   OF   A   TRIANGULATION. 

116.  The  adjustment  of  the  measured  angles   of  a  tri- 
angulation  net  is  a  special  case  of  the  problem  discussed 
in  the  preceding  chapters.     We   assume  the  reader  to  be 
acquainted  with  the  construction  and  method  of  handling 
of  instruments  used  in  measuring    horizontal    angles,    and 
shall   confine  ourselves  to  the    methods   of  adjusting   the 
measured  values  of  the  angles. 

117.  For  clearness  we  will  explain  in  some  detail  the 
preliminary  work  necessary  for  the  formation  of  the  con- 
dition   equations.     In   a   triangulation    there  must   be    one 
measured  base  at  least,  as  AB.     Starting  from  this  base  and 

measuring  the  angles  CAB,  ABC,  we  may 
compute  the  sides  AC,  BCbj  the  ordinary 
rules  of  trigonometry.  In  plotting  the 
figure  the  point  C  can  be  located  in  but 
one  way,  as  only  the  measurements  neces- 
F'9'17  sary  for  this  purpose  have  been  made. 

Similarly,  by  measuring  the  angles  CBD,  DCB  we  may 
plot  the  position  of  the  point  D,  and  this  can.be  done  in  but 
one  way.  If,  however,  the  observer,  while  at  A,  had  also 
read  the  angle  DAB,  then  the  point  D  could  have  been 
plotted  in  two  ways,  and  we  should  find  in  almost  all  cases 
that  the  lines  AD,  BD,  CD  would  not  intersect  in  the  same 
point.  In  other  words,  in  computing  the  length  of  a  side 
from  the  base  we  should  find  different  values,  according  to 
the  triangles  through  which  we  passed.  Thus  the  value  of 
CD  computed  from  AB  would  not,  in  general,  be  the  same 
if  found  from  the  triangles  ABC,  BCD,  and  from  ABC,  CAD. 


APPLICATION   TO   TRIANGULATION.  251 

If  the  blunt  angle  ABD  had  also  been  measured  we 
should  have  another  contradiction,  arising  from  the  non- 
satisfaction  of  the  relation 

DBC+  CBA  +  ABD  =  360° 

And  not  these  contradictions  only.  For  \ve  have  considered 
so  far  that  in  a  triangle  only  two  of  the  angles  are  meas- 
ured. If  in  the  first  triangle,  ABC,  the  third  angle,  BCA, 
were  also  measured,  we  know  from  spherical  geometry  that 
the  three  angles  should  satisfy  the  relation 

CA£  +  ABC+BCA  =  i8o0  +  sph.  excess  of  triangle 

which  the  measured  values  will  not  do  in  general.  A 
similar  discrepancy  may  be  expected  in  the  other  triangles. 
In  a  triangulation  net,  then,  with  a  single  measured 
base,  in  which  the  sides  are  to  be  computed  from  this  base 
through  the  intervening  triangles,  we  conclude  that  the 
contradictions  among  the  measured  angles  may  be  removed 
and  a  consistent  figure  obtained  if  the  angles  are  adjusted 
so  as  to  satisfy  the  two  classes  of  conditions: 

(1)  Those  arising  at  each  station  from   the  relations  of 
the  angles  to  one  another  at  that  station. 

These  are  known  as  local  conditions. 

(2)  Those  arising  from  the  geometrical  relations  neces- 
sary to  form  a  closed  figure. 

(a)  That  the  sum  of  the  angles  of  each  triangle  in  the 
figure  should   be  equal  to    180°   increased  by  the  spherical 
excess  of  the  triangle. 

(b)  That  the  length  of  any  side,  as  computed  from  the 
base,  should  be  the  same  whatever  route  is  chosen. 

These  are  known  as  general  conditions. 

118.  The  number  of  conditions  to  be  satisfied  will  de- 
pend on  the  measurements  made.  Each  condition  can  be 
stated  in  the  form  of  an  equation  in  which  the  most  prob- 
able values  of  the  measured  quantities  are  the  unknowns. 
The  number  of  equations  being  less  than  the  number  of 

33 


252  THE   ADJUSTMENT   OF   OBSERVATIONS. 

unknowns,  an  infinite  number  of  solutions  is  possible.  The 
problem  before  us  is  to  select  the  most  probable  values 
from  this  infinite  number  of  possible  values. 

The  general  statement  of  the  method  of  solution  is  this: 
Adjust  the  angles  so  as  to  satisfy  simultaneously  the  local 
and  general  conditions ;  that  is,  of  all  possible  systems  of 
corrections  to  the  observed  quantities  which  satisfy  these 
conditions,  to  find  that  system  which  makes  the  sum  of  the 
squares  of  the  corrections  a  minimum. 

The  form  of  the  reduction  depends  on  the  methods  em- 
ployed in  making  the  observations.  These 
methods,  in  general  terms,  are  as  follows: 
Let  O  in  the  figure  be  the  station  occupied, 
*  and  A,  B,  C  signals  sighted  at.  The  angles 
AOB,  BOC  are  required.  By  pointing  at  A 
and  then  at  B  we  find  the  angle  AOB. 
Point  now  at  B  and  next  at  C,  and  we  have 
the  angle  BOC.  These  two  angles  are  independent  of  one 
another. 

If,  however,  we  had  pointed  at  A,  B,  Cm  succession  we 
should  also  have  found  the  angles  AOB,  BOC,  but  they 
would  not  be  independent  of  one  another,  as  the  reading  to 
B  enters  into  each. 

In  the  first  method  of  measurement,  which  is  known  as 
the  method  of  independent  angles,  either  a  repeating  or  a 
non-repeating  theodolite  may  be  used ;  in  the  second,  or 
method  of  directions,  a  non-repeating  theodolite  only. 


The  Method  of  Independent  Angles. 

119.  As  the  case  of  independent  angles  is  the  simplest  to 
reduce,  we  shall  begin  with  it. 

A  distinction  must  be  made  between  angles  that  are  in- 
dependently observed  and  angles  which  are  independent  in 
the  sense  that  no  condition  exists  between  them.  Thus  at 
the  station  O,  above,  the  angles  AOB,  BOC,  AOC  might  be 


APPLICATION   TO   TRIANGULATION.  253 

observed  independently  of  one  another,  but  we  should  not 
call  them  independent  angles,  since  the  condition 

AOC=AOB  -\-BOC 

must  be  satisfied  between  them.  By  independent  angles, 
therefore,  in  the  reduction,  we  mean  those  measured  angles 
in  terms  of  which  all  the  measured  angles  can  be  expressed 
by  means  of  the  conditions  connecting  them.  In  the  pre- 
sent case  any  two  of  the  three  angles  AOB,  BOC,  AOC 
may  be  taken  as  independent,  and  the  third  angle  would  be 
dependent. 

Angles  may  be  measured  independently  either  with  a 
repeating  or  with  a  non-repeating  theodolite.  In  primary 
work  a  non-repeating  theodolite  in  which  the  graduated 
limb  is  read  by  microscopes  furnished  with  micrometers  is 
to  be  preferred.  The  method  of  reading  an  angle  is  as  fol- 
lows: The  instrument,  having  been  carefully  adjusted,  is 
directed  to  the  left-hand  signal  and  the  micrometers  read. 
It  is  then  directed  to  the  other  signal  and  the  micrometers 
again  read.  The  difference  between  these  readings  is  called 
a  positive  single  result.  The  whole  operation  is  repeated  in 
reverse  order;  that  is,  beginning  with  the  second  signal 
and  ending  with  the  first,  giving  a  negative  single  result. 
The  mean  of  these  two  results  is  called  a  combined  result, 
and  is  free  from  the  error  arising  from  uniform  twisting:  of 

cy  o 

the  post  or  tripod  on  which  the  instrument  is  placed,  or 
from  "  twist  of  station,"  as  it  is  called. 

The  telescope  is  next  turned  180°  in  azimuth  and  then 
1 80°  in  altitude,  leaving  the  same  pivots  in  the  same  wyes, 
and  another  combined  result  obtained.  The  mean  of  the 
two  combined  results  is  free  from  errors  of  the  instrument 
arising  from  imperfect  adjustments  for  collimation,  from  in- 
equality in  the  heights  of  the  wyes,  and  from  inequality  of 
the  pivots. 

The  distinction  between  these  two  combined  results  is 
noted  in  the  record  by  "  telescope  direct "  and  "  telescope 
reverse." 


254  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Besides  those  mentioned  there  are  two  kinds  of  system- 
atic error  in  measuring  angles  that  deserve  special  atten- 
tion. They  are  the  errors  arising  from  the  regular  or 
"  periodic  "  errors  of  graduation  of  the  horizontal  limb  of 
the  instrument,  and  the  error  from  the  inclination  of  the 
limb  itself  to  the  horizon.  The  effects  of  the  first  may  be 
got  rid  of  by  the  method  of  observation,  as  follows  :  The 
reading  of  the  limb  on  the  first  signal  is  changed  (usually 
after  each  pair  of  combined  results)  by  some  aliquot  part 
of  the  distance,  or  half-distance,  between  consecutive  micro- 
scopes in  case  of  two-microscope  and  three-microscope  in- 
struments respectively.  Thus  if  n  is  the  number  of  pairs  of 

combined  results  desired,  the  changes  would  be  —  and  - 

n  n 

respectively  with  the  instruments  mentioned.     The  opera- 
tion of  reversal  in  case  of  a  three-micro- 
scope instrument  causes  each  microscope 
to  fall  at  the  middle  of  the  opposite  120° 
space,    the    limb    remaining    unchanged. 
Thus  if  the  full  lines  in  Fig.  18  represent 
the  positions  of  the  microscopes  with  tele- 
scope direct,  the  dotted  lines  show  their 
positions  with  telescope  reverse.      In  this 
lies  the  greatest  advantage  of  three  micro- 
scopes  over  two,  since  with    the   latter,  in  reversing,  the 
microscopes  simply  change  places  with  each  other,  without 
reading  on  new  portions  of  the  limb. 

The  error  arising  from  want  of  level  of  the  horizontal 
limb  cannot  be  eliminated  by  the  method  of  observation,  but 
with  the  levels  which  accompany  a  good  instrument,  and 
with  ordinary  care,  it  will  usually  be  less  than  o".i.  In  case, 
however,  of  a  signal  having  a  high  altitude  above  the  hori- 
zon, the  error  from  this  source  may  be  greater,  and  then 
special  care  should  be  taken  in  levelling.  For  an  expres- 
sion for  its  influence  in  any  case  see  Chauvenet's  Astronomy, 
Vol.  II.  Art.  211. 

The  observations  should  be  made  on  at  least  two  days 


APPLICATION   TO   TRIANGULATION. 


255 


when  conditions  are  favorable.  Results  obtained  at  differ- 
ent hours  of  a  day  are  of  more  value  than  the  same  number 
of  results  obtained  on  different  days  at  the  same  hour  of  the 
day.  This  is  on  account  of  variation  in  external  conditions 
(direction  of  light,  phase,  distinctness,  refraction,  etc.) 

120.  We  shall  for  illustration  take  the  following  example,  making  use  of 
such  parts  of  it  from  time  to  time  as  may  belong  to  the  subject  in  hand,  and 
finally,  after  explaining  the  method  of  forming  the  condition  equations, 
solve  it  in  full. 

In  the  triangulation  of  Lake  Superior  executed  by  the  U.  S.  Engineers 
the  following  angles  were  measured  in  the  quadrilateral  N.  Base,  S.  Base, 
Lester,  Oneota. 


LNO  = 

124° 

09' 

40". 

69     wei: 

SNL  = 

"3° 

39' 

05". 

07          ' 

ONS  = 

122° 

li' 

15" 

.61 

NSO  = 

23° 

08' 

05", 

.26 

LSN  = 

47° 

3l' 

20". 

4i 

LSO  = 

70° 

39' 

24", 

,60          ' 

SON  = 

34° 

40' 

39" 

,66 

NOL  - 

43° 

46' 

26" 

.40 

OLS  = 

30° 

53' 

30" 

,81          « 

These  angles  we 

shall 

denote  by  Mi, 

Fig.19 


14 

23 

6 

7 

3i 
i 
8 


..  M»  respectively. 


The  length  of  the  line  N.  Base  —  S.  Base  (Minnesota  Point)  is  6o56"'.6, 
and  the  latitudes  of  the  four  stations  are  approximately 


N.  Base,  46°  45' 
S.  Base,    46°  43' 


Lester,     46°  52' 
Oneota,   46°  45' 


121.  The  Local  Adjustment. — When  in  a  system  of 
triangulation  the  horizontal  angles  read  at  a  station  are 
adjusted  for  all  of  the  conditions  existing  among  them,  then 
these  angles  are  said  to  be  locally  adjusted. 

From  the  considerations  set  forth  in  Art.  1 17  it  is  readily 
seen  that  at  a  station  only  two  kinds  of  conditions  are 
possible— 

(a)  that   an   angle    can    be   formed    from    two   or   more 
others,  and 

(b)  that  the  sum  of  the  angles  round  the  horizon  should 
be  equal  to  360°. 


256  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  second  of  these  is  included  in  the  first,  and  the 
method  of  adjustment  may  be  stated  in  general  terms  as 
follows  : 

An  inspection  of  the  figure  representing  the  angles  at 
the  station  will  show  how  all  of  the  measured  angles  can  be 
expressed  in  terms  of  a  certain  number  of  them  which  are 
independent  of  one  another.  These  relations  will  give  rise 
to  condition  equations,  or  local  equations,  as  they  are  called, 
which  may  be  solved  as  in  Chapters  IV.  or  V. 

Thus,  if  M,,  Mv  .  .  .  Mn  denote  the  single  measured 
angles,  and  •?',,  vv  .  .  .  vn  their  most  probable  corrections, 
then  if  any  of  the  angles  Mh,  Mk  can  be  formed  from  others, 
we  have,  by  equating  the  measured  and  computed  values, 
the  local  condition  equations, 


or 

<',  +  ?'„  +  •   •   •  —  *'*  =  4  suppose 


with 

'  =  a  minimum 


where  A>  A>  •   •   •  A  denote  the  weights  of  the  angles. 

The  solution  ma)?  be  in  general  best  carried  out  by  the 
method  of  correlates,  as  in  Chap.  V. 

The  following  special  cases  are  of 
frequent  occurrence  : 

(i)  At  a  station  O  the  n  —  i  single 
angles  AOB,  BOC,  .  .  .  are  measured, 
and  also  the  sum  angle  AOL,  to  find  the 
adjusted  values  of  the  separate  angles, 
all  of  the  measured  values  being  of  the 
same  weight. 


APPLICATION   TO   TRIANGULATION.  257 

The  condition  equation  is 


or 


=  /  suppose, 
with 

[V*]  =  a  minimum. 

The  solution  gives  (Art.  109  or  no) 


that  is,  the  correction  to  each  angle  is  -  of  the  excess  of  the  sum 

angle  over  the  sum  of  the  single  angles,  and  the  sign  of  the  cor- 
rection to  the  sum  angle  is  opposite  to  that  of  the  single  angles. 

(2)  At  a  station  O  the  n  single  angles  AOB,  BOC,  .  .  . 
LOA  are  measured,  thus  closing  the  horizon,  to  find  the 
adjusted  values  of  the  angles. 

The  condition  equation  is 


=  I  suppose, 
with  [/#a]  =  a  minimum. 

The  solution  gives 


where  «,  =  -,«,  =  -,.    .  .,  and  [//]  =  I  . 

A          A  L/J 

If  the  weights  are  equal,  then 

l_ 

^  —  v,—  .  .  .  —  ?'„  -  — 

that  is,  the  correction  to  each  angle  is  -   of  the  excess  of  360° 
over  the  sum  of  the  measured  angles. 


258  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  i.  The   angles   at  station   N.  Base  close   the   horizon  ;    required   to 
adjust  them. 

We  have  (Art.  120) 

M-i  +  ?'i  =  124°  09'  40".  69  +  v-i     weight  2 
Mi  +  vt  =  113°  39'  05".  07  +  v-t  "       2 

M3  +  z'3  =  i22°  n'  15".  61  +  z>3  "      14 

Sum        =  360°  oo'  oi".37  +  v\  +  v?  +  vs 
Theoretical  sum        =  360°  oo'  oo".oo 

.'.   Local  equation  is  o     =      i".37  +  v\  +  v*  +  va 

Hence  (Ex.  2,  Art.  no) 


=  —  o".64 
z>z  =  —  o".64 
Z'a  =  —  o".O9 
and  the  adjusted  angles  are 

124°  09'  40".  05 
113°  39'  04".  43 

122°    II'    15".  52 

Check-sum  =  360°  oo'  oo".oo 

Ex.  2.  Precisely  as  in  the  preceding  we  may  deduce  at  station  South  Base 
the  local  equation 

o  =  i".o7  +  vt  +  v6  —  v6 
and  the  adjusted  angles 

23°  08'  05".  13 
47°  31'  19".  91 
70°  39'  2  5  ".04 

122.  Number  of  Local  Equations  at  a  Station.  —  If  s  sta- 
tions are  sighted  at  from  a  station  that  is  occupied,  the 
number  of  angles  necessary  to  be  measured  to  determine  all 
of  the  angles  that  can  be  formed  at  the  station  occupied  is 
s—i.  If,  therefore,  an  additional  angle  were  measured, 
its  value  could  be  determined  in  two  ways  :  from  the  direct 
measurement  and  from  the  s  —  i  measures.  The  contradic- 
tion in  these  two  values  would  give  rise  to  a  local  (condition) 
equation.  If,  therefore,  n  is  the  total  number  of  angles 
measured  at  a  station,  the  number  of  local  equations,  as 
indicated  by  the  number  of  superfluous  angles,  is 

n  —  s-\-  I. 


APPLICATION   TO   TRIANGULATION.  259 

123.  The     General    Adjustment.  —  With    a    single 
measured  base  the  number  of  conditions  arising  from  the 
geometrical  relations  existing  among  the  different  parts  of 
a  triangulation  net  can  be  readily  estimated.     For  if  the  net 
contains  s  stations,  two  are  known,  being  the  end  points  of 
the  base,  and  s  —  2  are  to  be  found. 

Now,  two  angles  observed  at  the  end  points  of  the  base 
will  determine  a  third  point ;  two  more  observed  at  the  end 
points  of  a  line  joining  any  two  of  these  points  will  de- 
termine a  fourth  point,  and.  so  on.  Hence  to  determine 
the  s  —  2  points,  2  (s  —  2)  angles  are  necessary.  If,  there- 
fore, ;/  is  the  total  number  of  angles  measured,  the  number 
of  superfluous  angles,  that  is,  the  number  of  conditions  to 
be  satisfied,  is 

n  —  2(s  —  2) 

Ex.  In  a  chain  of  triangles,  if  s  is  the  number  of  stations,  show  that  the 
number  of  conditions  to  be  satisfied  is  s  —  2  ;  and  in  a  chain  of  quadrilaterals, 
with  both  diagonals  drawn,  the  number  of  conditions  is  2s  —  4. 

The  equations  arising  from  these  conditions  are  divided 
into  two  classes,  angle  equations  and  side  equations. 

124.  The  Angle  Equations. — -The  sum  of  the  angles  of  a 
triangle  drawn  on  a  plane  surface  is  equal  to  180°.     The 
sum  of  the  angles  of  a  spherical  triangle  exceeds  180°  by  the 
spherical  excess  (e)  of  the  triangle,  which  latter  is  found 
from  the  relation 

_area  of  triangle 
R  sin  \" 

R  being  the  radius  of  the  sphere. 

From  surveys  carried  on  during  the  past  two  centuries 
the  earth  has  been  found  to  be  spheroidal  in  form,  and  its 
dimensions  have  been  determined  within  small  limits.  Now, 
a  spheroidal  triangle  of  moderate  size  may  be  computed  as 
a  spherical  triangle  on  a  tangent  sphere  whose  radius  is 
Vpl  ftt,  where  />„  py  are  the  radii  of  curvature  of  the  meridian 
and  of  the  normal  section  to  the  meridian  respectively  at 

34 


260 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


the  point  corresponding-  to  the  mean  of  the  latitudes  tp  of 
the  triangle  vertices. 

Hence  we  may  wrap  our  triangulation  on  the  spheroid 
in  question  by  conforming  it  to  the  spherical  excess  com- 
puted from  the  formula 

rt^sin  C 

£  (in  seconds)  = : 7. 

2,o1/>2  sin  i 

where  a,  b  are  two  sides  and  C  is  the  included  angle  of  the 
triangle. 

For  convenience  of  computation  we  may  write 

€  =  A  ab  sin  C 

when  log  A  may  be  tabulated  for  the  argument  <p.  The  fol- 
lowing table  is  computed  with  Clarke's  values  of  the  ele- 
ments of  the  terrestrial  spheroid  corresponding  to  latitudes 
from  10°  to  70°.  The  metre  is  the  unit  of  length  to  be 
used. 


V 

log  A 

<P 

log  A 

9 

log  X 

10° 

1.40675 

30° 

1.40547 

50° 

1.40349 

11° 

672 

3i° 

537 

5i° 

339 

12° 

668 

32° 

528 

52° 

329 

13° 

663 

33° 

519 

53° 

319 

14° 

659 

34° 

509 

54° 

309 

15° 

1.40654 

35° 

1.40500 

55° 

1.40299 

16° 

649 

36° 

49  1 

56° 

289 

17° 

643 

37° 

481 

57° 

280 

18° 

637 

38° 

47i 

58° 

271 

19° 

631 

39° 

461 

59° 

262 

20° 

1.40625 

40° 

1.40451 

60° 

1.40253 

21° 

618 

4i° 

441 

61° 

244 

22° 

611 

42° 

43i 

62° 

235 

23° 

604 

43° 

420 

63° 

226 

24° 

597 

44° 

410 

64° 

218 

25° 

1.40589 

45° 

1.40400 

65° 

1.40210 

26° 

58i 

46° 

39° 

66° 

202 

27° 

573 

47° 

380 

67° 

195 

28° 

564 

48° 

369 

68° 

188 

29° 

555 

49° 

359 

69° 

181 

30° 

1-40547 

50° 

1.40349 

70° 

1.40174 

APPLICATION   TO   TRIANGULATION.  26 1 

To  find  a,  b,  tp,  a  preliminary  geodetic  computation  must 
first  be  made  of  the  triangulation  to  be  adjusted,  starting 
from  a  base  or  from  a  known  side.  The  values  found  from 
using  the  unadjusted  angles  will  be  close  enough  for  this 
purpose.  The  latitudes  need  only  be  computed  to  the 
nearest  minute. 

A  useful  check  of  the  excess  results  from  the  principle 
that  the  sums  of  the  excesses  of  triangles  that  cover  the 
same  area  should  be  equal.  In  our  example  the  spherical 
excesses  of  the  triangles  ONS,  LSO  will  be  found  to  be 
o".o5  and  o".37  respectively. 

In  each  single  triangle,  then,  the  condition  required  to 
wrap  it  on  the  spheroid,  that  is,  that  the  sum  of  the  three 
measured  angles  shall  be  equal  to  180°,  together  with  the 
spherical  excess,  gives  a  condition  equation.*  This  is 
called  an  angle  equation,  or  by  some  a  triangle  equation. 

Ex.  In  the  triangle  N.  Base,  S.  Base,  Oneota,  if  573,  #4,  v*  denote  the  cor- 
rections to  the  three  angles,  we  have  for  the  most  probable  values 

ONS=.  122°  n'  15". 61  +  z>3 
NSO=  23°  08'  05". 26  +  z>4 
SON  =  34°  40'  39". 66  +  z>7 

Sum  =  180°  oo'  oo".53  +  z»3  +  z'4  +  z-7 
Theoretical  sum  =  180°  oo'  oo".o5  =  180°+  s 

and  the  angle  equation  is  formed  by  equating  these  sums.     The  result  is 

<^3  +  v\  +  vi  +  o".4S  =  o 
Similarly  from  the  triangle  Lester,  S.  Base,  Oneota,  the  angle  equation  is 

Ve  +  Z'7  +  V6  +  "'a  +  I*.  10  =  0 

125.  Number  of  Angle  Equations  in  a  Net. — It  is  to  be 
expected  that  in  a  triangulation  net  some  of  the  lines  will 
be  sighted  over  in  both  directions,  and  some  in  only  one 
direction.  If  these  latter  lines  are  omitted  the  number  of 
angle  equations  will  remain  unaltered.  Thus  in  our  Lake 
Superior  quadrilateral  (Fig.  19)  the  line  NL  has  been 

*  We  confine  ourselves  throughout  to  triangles  to  which  Legendre's  theorem  is  applicable. 
For  very  large  triangles  other  formulas  for  spherical  excess  must  be  used.  See,  for  example, 
Helmert,  Theorieen  d.  lioheren  Gcoddsie,  vol.  i.  p.  362. 


262  THE  ADJUSTMENT   OF   OBSERVATIONS. 

sighted  over  from  N,  but  not  from  L,  so  that  we  have  only 

two  angle  equations:  namely,  those 
resulting  from  the  triangles  ONS, 
OLS,  just  as  if  the  figure  had  been 
Of  the  form  of  Fig.  21,  in  which  the 
line  NL  is  omitted. 

Generally,  if  s  is  the  number  of 
stations  occupied,  the  polygon  form- 
ing the  outline  of  the  net  will  give 
rise  to  one  angle  equation.  Each 
diagonal  that  is  drawn  will  form  a 
figure,  giving  rise  to  an  additional 
angle  equation.  Hence  if  in  the  net  there  are  /t  lines 
sighted  over  in  both  directions,  the  number  of  diagonals 
will  be  /,  —  s,  and  the  number  of  angle  equations 

Z-t+l 

If  /2  of  the  lines  are  sighted  over  in  one  direction  only, 
and  /  is  the  total  number  of  lines  in  the  figure,  then  since 
/,=/  — /2,  the  number  of  angle  equations  would  be  ex- 
pressed by 

l-l-s+i 

Thus  in  the  figure  the  polygon  LONS  gives  an  angle 
equation,  and  the  line  OS  gives  rise  to  a  second  angle  equa- 
tion from  either  the  triangle  ONS  or  OLS.  We  might, 
therefore,  form  the  equations  from  either  of  the  three  sets 
of  figures, 

LONS  LONS  ONS 

ONS  OLS  OLS 

and  should  have  respectively 

»!  +  «'.  +  «'.  +  «'•  +  »•+ 3-06=0 

^3  +  ^  +  ^+0.48  =  0 

^  +  v*  +  ^  +  vs  +  vt  +  3.06  =  o 

VK  +  V7-\-Vs  +  V,+    1-10=0 

^3  +  v<  +  v,  +  0.48  =  o 

»         »         W         »         *•  *0  =  O 


APPLICATION   TO   TRIANGULATIOX. 


263 


which  pairs  of  equations,  by  means  of  the  relations  already 
found  (pp.  258,  261), 

*'!  +  *'•  +  *'•+ 1-37  =  0 

W4  +  V.  —  V.+ 1.07=0 


reduce  to  the  same  two  equations  for  each  set  of  polygons. 

Ex.  In  the  quadrilateral  ABCD,  in  which  all  of  the  8  angles  are  measured, 
show  that  there  are  three  independent  angle  equations, 
and  that  these  equations  may  be  found  from  the  follow-     A 
ing  8  sets  of  figures  : 

ABD,  ABC,  A  CD;  ABD,  ABC,  ABCD; 
ABD,  A  CD,  ABCD; 

BDA,  BCD,  BCA  ;  BCA,  BCD,  BCD  A  ; 
CDB,  CAB,  CD  A ;  CDB,  CD  A,  CDBA; 
DAB,  DBC,  DAC. 


Fig. 22 


Fig.23 


126.  The  Side  Equations. — In  a  single  triangle,  or  in  a 
simple  chain  of  triangles,  the  length  of  any  assigned  side 
can  be  computed  from  a  given  side  in  but  one  way.  When 
the  triangles  are  interlaced  this  is  not  the  case. 

Thus  in  Fig.  21  any  side  can  be  computed  from  ATS  in 
but  one  way.  The  only  condition  equations  apart  from  the 

,Li  local  equations  would  be  the  two 
angle  equations.  But  in  Fig.  19, 
in  which  the  line  NL  is  sighted 
over  from  N,  we  have  the  further 
condition  that  the  lines  OL,  NL, 
SL  intersect  in  the  same  point,  L. 
The  figure  plotted  from  the  meas- 
ured values  would  be  of  the  form 
of  Fig.  23. 

To  express  in  the  form  of  an 
equation  the  condition  that  the 
three  points  L,,  L.,,  7,3  must  coincide,  we  proceed  as  follows: 
Starting  from  the  base  ATS,  we  may  compute  SL,  directly 


264  THE   ADJUSTMENT   OF   OBSERVATIONS. 

from  the  triangle  SNLl  and  SLS  from  the  triangles  SON, 
SOLy     This  gives 

sin  SN  _  sin  SL,N 


sin  SL1       sin 

sin  SN  _  sin  SON  sin  SL3O 
sin  SL3  ~~  sin  SNO   sin  SOL3 

But  SL,  must  be  equal  to  S£3 

Hence  the  condition  equation  is 

sin  SLN     sin  SNO     sin  5(97. 


sTn~52VX     sin  SON    sin 
which  is  called  a  .rzV&  equation  or  5//^  equation. 


Ex.   In  the  figure  ABCD3  Di  A  ,  the  three  angle  equations, 

BD^A  =  180° 


°3  "  "^    +  -ffC/4    +  CAB    =  180°  + 


Fi9-24  .#CZ>3   +  CZ>3J5  +  DzBC  =  1 80°  +  £3 

D, 

given  by  the  triangles  D^AB,  ABC,  BCD*,  may  be  satis- 
fied and  yet  the  figure  not  be  a  perfect  quadrilateral. 
Show  by  equating  the  values  of  BDi  and  BD$  that  the 
further  condition  necessary  is 

sin  DAB  sin  BCA  sin  CDB  _ 
sin  BDA  sin  CAB  sin  BCD 

The  side  equation 

sin  SLN    sin  SOL     sin  .S7V0  _ 


sin  SNL     sin  SZ<9     sin  S6W 
gives  the  identical  relation 

sin  SN    sin  SL     sin  50 


sin  >SX      sin  SO     sin  .S/V 


_ 


Hence  in  forming  a  side  equation  we  may  proceed  mechani- 
cally in  this  way.     Write  down  the  scheme 

SN    SL     50_ 
SL     SO    SN~l 


APPLICATION   TO   TRIANGULATION.  265 

the  numerator  and  denominator  each. being  formed  by  the 
lines  radiating  from  the  point  S  in  order  of  azimuth,  and  the 
first  denominator  being  the  second  numerator.  The  side 
equation  results  from  replacing  the  sides  by  the  sines  of  the 
angles  opposite  to  them. 

The  point  5  is  called  the/<?<fc  of  the  quadrilateral  for  this 
equation. 

127.  Position  of  Pole. — It  is  easily  seen  that  in  forming  the 
side  equation  any  vertex  may  be  taken  as  pole.  For  plot- 
ting the  figure  from  the  angles  of  the  triangles  ONS,  OLS, 
the  side  equation  with  pole  at  S  means  that  the  points  Lt 
and  L3  must  coincide.  The  side  equation  with  pole  at  N 
means  that  Z-,,  L^  coincide,  and  with  pole  at  O  that  L^  L3 
coincide.  If  any  one  of  these  conditions  is  satisfied  the 
others  are  also  satisfied,  as  each  amounts  to  the  same  con- 
dition that  L  is  not  three  points  but  one  point. 

Similar  reasoning  will  show  that  by  plotting  the  figure 
from  LONS,  ONS,  the  side  equations  formed  by  taking  the 
poles  at  N,  L,  S,  mean  that  O  is  not  three  points  but  one 
point,  and  so  on.  Hence  the  side  equation  formed  from 
any  vertex  as  pole  in  connection  with  the  angle  equations 
fixes  each  point  of  the  figure  definitely  and  removes  all 
contradictions  from  it. 

It  will  be  noticed  that  the  reasoning  is  in  no  way  affected 
by  the  line  NL  being  sighted  over  in  only  one  direction. 

Ex.  i.  In  a  quadrilateral  ABCD,  in  which  all  of  the  8  angles  are  measured, 
show  that  of  the  15  side  equations  that  may  be  formed,  7  only  are  different  in 
form,  and  that  by  taking  the  angle  equations  into  account  all  of  them  may 
be  reduced  to  a  single  form. 

Also  show  that  there  are  56  ways  of  expressing  the  three  angle  and  one 
side  equations  necessary  to  determine  the  quadrilateral. 

Ex.  2.  Examine  the  truth  of  the  following  statement.  In  a  quadrilateral 
an  angle  equation  may  be  replaced  by  a  side  equation,  so  that  the  quadri- 
lateral may  be  determined  by  3  angle  equations  and  one  side  equation,  2 
angle  equations  and  2  side  equations,  one  angle  equation  and  3  side  equa- 
tions, the  number  of  conditions  remaining  four,  and  the  four  not  being  all  of 
one  kind. 


266 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


128.  If  the  triangulation  net,  instead  of  involving  quad- 
rilaterals only,  involves  central  polygons,  such  that,  in  com- 
puting the  lengths  of  the  sides, 
we  can  pass  from  one  side  to 
any  other  through  a  chain  of 
triangles,  the  same  process  is 
followed  in  forming  the  side 
equations  as  in  a  quadrilateral. 
Thus  in  the  figure  which  rep- 
resents part  of  the  triangula- 
tion of  Lake  Erie  west  of  Buf- 
falo Base  there  are  side  equa- 
tions from 


Fifl.25 


the  quadrilaterals  CDHG,  GHFA 
the  pentagons  GABCH,  HGDEF 

The  scheme  for  the  pentagonal  side  equation  GABCH,  for 
example,  would  be  just  as  in  the  case  of  a  central  quadri- 
lateral, taking  G  as  pole, 


GA 
GB 


GB_ 
~GC 


and  the  side  equation 

sin  GBA     sin  GCB 


GC      GH  _ 
~GH     "GA  ~ 


sin  GHC    sin  GAH 


sin  GAB    sin  GBC    sin  GCH    sin  GHA 


=  i 


129.  Reduction  to  the  Linear  Form.  —  Thus  far  we  have 
considered  the  side  equations  in  their  rigorous  form.  But 
in  order  to  carry  through  the  solution  by  combining  them 
with  the  other  condition  equations  they  must  be  reduced  to 
the  linear  form.  We  proceed  to  show  how  this  may  be 
done.  (See  Art.  7.) 

Let  the  side  equation  be 

sin  F,     sin  F3 
~ 


where  F,,  F2,  . 
angles.     Let 


.  denote  the   most  probable  values  of  the 
Mv  .  .  .  denote  the  measured  values,  and 


APPLICATION   TO   TRIANGULATION.  267 

vt,  vv  .  .  .  the  most  probable  corrections  to  these  values  ; 
then  the  equation  may  be  written 

sin  (Ml  -\-  Q     sin  (Mt  -f  vt) 
sin  (M,  +  z/a)     sin  (Mt  +  v4)  ' 

Taking  the  log.  of  each  side  of  this  equation  and  expanding 
by  Taylor's  theorem,  we  have,  retaining  the  first  powers  of 
the  corrections  only, 

log  sin  M,  +  -JM  (log  sin  M,)  v, 

f  .  .   .=o    (3) 


which  may  be  written  in  two  forms  for  computation  : 

First,  if  the  corrections  to  the  angles  are  expressed  in 
seconds,  we  may  put 


where  d'  is  the  tabular  difference  for  \"  for  the  angle  M1  in  a 
table  of  log.  sines.     Then  we  have 

d'v,  —  o'v,  -f  .  .  .  +  log  sin  M,  —  log  sin  M,  -f  .  .  .  =  o 

that  is, 

[oV]  =  /  (4) 

where  /  is  a  known  quantity. 
Secondly,  we  may  replace 

-r^r  (log  sin^f,)     by     mod  sin  \"  cot  J/, 

where  mod  denotes  the  modulus  of  the  common  system  of 
logarithms.     Eq.  3  may  then  be  arranged 

Cot  M,  v,  —  cot  M,  v9  +  .  .  . 

=  —,  -  ^    -•  -  -  (log  sin  M.  —  log  sin  M.  +  .  .  .)    (5) 
10'  mod  sin  i  ' 

if  the  seventh  place  of  decimals  is  chosen  as  the  unit. 

35 


268  THE   ADJUSTMENT   OF   OBSERVATIONS. 

For  convenience  of  computation  there  is  not  much  to 
choose  between  the  two  forms.  The  second  is  perhaps,  on 
the  whole,  to  be  preferred  in  ordinary  triangulation  work 
with  well-shaped  triangles. 

For  the  method  of  computing  log.  sines  and  log.  differ- 
ences for  small  angles  or  for  angles  near  180°,  and  also  if  a 
ten-place  table  is  used,  see  Art.  7. 

130.  Check  Computation.  —  The  side  equation  deduced 
from  spherical  triangles  must  also  follow  from  the  corre- 
sponding plane  triangles,  the  angles  of  each  spherical  tri- 
angle being  transformed  according  to  Legendre's  theorem  ; 
that  is,  for  example,  we  should  obtain  the  same  constant 
term  /  by  reducing  to  the  linear  form  the  equation 

sin  SLN    sin  SOL     sin  SNO  __ 
sin  SNL     sin  SLO    sin  SON~ 

or  the  equation 

s'm(SLN-^)     s'in(SOL-^)    sin  (SNO  -  % 


sn 


_ 
~ 


where  s19  £2,  «3  denote  the  spherical  excesses  of  the  triangles 
SNL,  SOL,  and  SON  respectively. 

It  affords  a  check  of  the  accuracy  of  the  numerical  work 
to  compute  the  side  equation  with  both  the  spherical 
angles  and  the  plane  angles.  It  is  evidently  simpler  to  use 
the  spherical  angles,  so  that  if  but  a  single  computation 
is  to  be  made  they  should  be  chosen. 

For  a  check  of  the  coefficients  of  the  corrections  we 
have,  by  expanding  the  second  equation  by  Taylor's  theorem, 
the  relations 

etf'  -  <T)  -f  fa(d'"  -  <T")  -f  £3(d"'"  -  <5""")  =  o 
or 

f,(cot  SLN—  cot  SNL)  -f  fa(cot  SOL  -  cot  SLO) 

+  £3(cot  SNO  -  cot  SON)  =  o 

for  the  first  and  second  forms  of  reduction  respectively. 
This  useful  check  is  given  by  Andrse. 


APPLICATION   TO   TRIANGULATION.  269 

Ex.  The  quadrilateral  N.  Base,  S.  Base,  Oneota,  Lester  (Fig.  19). 
Take  the  pole  at  Lester. 
We  have  the  scheme 

LS  LN  LO  _ 

LN  LO~LS~ 

from  which  we  write  down  the  side  equation 

sin  LNS  sin  LON  sin  LSO  _ 
sin  LSN  sin  LNO  sin  LOS  ~ 
that  is, 

sin  (Mi  +  7-a)  sin  (Ma  +  7'fi)  sin  (Me  +  VK) 


sin  (Hf.,  +  z-s,)  sin  (A/i  +  7-1)  sin  (M-,  +  Ms  +  7-,  +  7-6) 

First  Form  of  Reduction. 

log  sin  (113°  39'  05". 07  +  7-2)  =9.9618969,7—    9,227-2 

log  sin  (  43°  46'  26". 40  +  7',)  =  9.8399903,4  +  21,98  7-g 

log  sin  (  70°  39'  24". 60  +  7-6)  =9.9747656,9+    7,397-6 

530,0 

log  sin  (  473  31'  20". 41  +  7-5)  =9.8677859,5  +  19,287-5 

log  sin  (124°  09'  40". 69  +  7-,)  =9.9177470,2—  14,297-! 

log  sin  (  78"  27'  06". 06  +  7-7  +  z-b)  =  9.991 1 180,3  +    4,30(7-7  +  rv) 

510,0 
Hence  the  side  equation  in  the  linear  form  is 

14.297-,  —  9.227-.J  —  19.287-5  +  7.397-6  — 4.307-7  +  17-687-8  +  20.0  =  0 
the  unit  being  the  seventh  place  of  decimals. 

Check  of  the  constant   term  by  computing  the  log.  sines  after  deducting 
from  each  angle  \  of  the  spherical  excess  of  the  triangle  to  which  it  belongs. 


Angle. 

113^  39'  05".  oo 
43'  46'  26  ".36 
"o°  39'  -4"-  4S 

Log  Sin. 

9.9618970,3 
9.8399902,5 

Angle. 

47^  31'  2o".34 
124'  09'  40".  65 
78    27'  05  ".93 

Log  Sin. 
9.8677858,2 
9.9177470,7 
9.9911179,8 

528,8 

508,7 

+  20,1 

agreeing  closely  with  the  value  found  from  the  spherical  angles. 
Check  of  the  coefficients. 
o.i9(—  9.22  —  19.28)  +  0.12(21.98  +  14.29)  +  0.37(7.39  —  4-3°)  =  +  o-oS 

This  check  would   have  been  closer  had  the  spherical  excesses  been  carried 
out  to  three  places  of  decimals.     We  have  taken  o'.i9,  o'.os,  and  o".i2  for  the 


2/O  THE  ADJUSTMENT   OF   OBSERVATIONS. 

excesses  of  the  single  triangles  LNS,    ONS,  LNO,  and  o".37  for  the  sum 
triangle  LSO.     The  first  two  are  more  nearly  o".ig5  and  o".o55. 

Second  Form  of  Reduction. 

Log  Sin.  Log  Sin. 

9.9618969,7  —  0.43802/2  9-8677859,5  +  0.91562/5 

9.8399903,4+  1.04372/8  9.9177470,2  —  0.67862/1 

9.9747656,9  +  0.35102/6  9.9911180,3  +  0.2043(777  +  t'») 

530,0 
510,0 


20,0      log    I.30IO3 

— j    log  8.67664 
io7  mod  sin  i 

9.97767      0.950 
and  the  side  equation  is 
0.67862/1  —  0.43802/2  —0.91562/5  +  0.35102/0  —  0.20432/7  +  0.83942/8  +  0.950  =  o 

This  result  may  be  checked  in  the  same  way  as  in  the  first  form. 

In  reducing  a  side  equation  to  the  linear  form  the  coefficients  of  the  cor- 
rections should  be  carried  out  to  one  place  of  decimals  farther  than  the  abso- 
lute term.  This  for  a  short  computation  would  be  unnecessary,  but  in  the 
reduction  of  an  extensive  triangulation  net  it  is  rendered  necessary  by  the 
accumulation  of  errors  from  the  dropping  of  the  last  figures  in  products  and 
quotients. 

It  will  be  noticed  that  in  the  preceding  example  we  have 
carried  out  the  log.  sines  to  8  places  of  decimals,  the  seventh 
place  being  the  unit.  This  is  amply  sufficient,  in  primary 
work,  for  our  present  methods  of  measurement,  as  already 
indicated  in  Art.  34.  Indeed,  it  is  in  general  sufficient  to 
carry  them  to  7  places  of  decimals  only,  an  ordinary  7-place 
table  being  used.  As  there  is  no  8-place  table  published, 
the  labor  of  forming  the  log.  sines  with  a  lo-place  table,  and 
then  cutting  down  the  results  to  8  places,  is,  in  very  many 
cases,  hardly  justified  by  the  extra  precision  attained. 

For  a  secondary  triangulation  6  places  of  decimals,  and 
for  a  tertiary  triangulation  5  places,  may  be  used.* 

*  On  the  Coast  Survey  and  Lake  Survey  the  practice  in  primary  triangulation  has  been  to 
carry  out  the  log.  sines  to  io  places  of  decimals.  On  the  English  Ordnance  Survey  they  were  also 
carried  out  to  io  places,  but  on  the  more  modern  Great  Trigonometrical  Survey  of  India  to  7 
places  only.  In  the  triangulation  of  Denmark,  Andra:  used  8  places  ;  and  in  this  he  has  been  fol- 
lowed by  the  Prussian,  Italian,  and  other  modern  European  surveys. 


APPLICATION   TO   TRIANGULATION.  2/1 

131.  We  have  seen  that  the  coefficients  of  the  correc- 
tions in  a  side  equation  are  given   by  the    differences   for 
i"  of  the  log.  sines  of  the  angles,  or  by  the  cotangents  of  the 
angles  that  enter.     Now,  since  an  equable  distribution  of 
errors  arising  from  the  approximate  computation   is   best 
attained   by  securing  the  greatest  possible  equality  of  co- 
efficients throughout  the  condition  equations,  and  since  the 
coefficients  of  the  corrections  in  the  angle  equations   are 
-f- 1  or  —  i,  it  follows  that  it  would  be  most  convenient  to 
put  the  side  equations  on  the  same  footing  as  the  angle 
equations.     To  do  this  we  may  divide  the  side  equation  by 
such  a  number  as  will  make  the  average  value  of  the  co- 
efficients equal  to  unity.     This,  for  angles  ordinarily  met 
with  in  triangulation,  would  be  effected  by  taking  the  sixth 
place  of  decimals  as  the  unit  in  the  side  equation.     Thus  in 
our  example,  dividing  by    10,  which  is  approximately  the 
mean  of  the  coefficients,  and  which  amounts  to  the   same 
thing  as  expressing  the  log.  differences  in  units  of  the  sixth 
place  of  decimals,  the  equation  may  be  written 

1.43^-0.92?',-  1.93^  +  0.74^-0.43^  +  i.777/8  +2.00  =  0 

It  would  have  been  equally  correct  to  have  multiplied 
each  of  the  angle  equations  by  10,  and  so  have  put  them  on 
the  same  footing  as  the  side  equations.  Dividing  the  side 
equations  is,  however,  to  be  preferred,  as  the  coefficients 
are  made  smaller  throughout,  and  the  formation  and  solu- 
tion of  the  normal  equations  is  consequently  easier. 

A  striking  difference  between  condition  equations  and 
observation  equations  is  here  brought  out.  As  a  condition 
equation  expresses  a  rigorous  relation  among  the  observed 
quantities  altogether  independent  of  observation,  it  may  be 
multiplied  or  divided  by  any  number  without  affecting  that 
relation ;  with  an  observation  equation,  on  the  other  hand, 
the  effect  would  be  to  increase  or  diminish  its  weight. 
(Compare  Art.  58.) 

132.  Position  of  Pole. —  In  a  quadrilateral,  taking  any  of  the 
vertices  as  pole,  the  conclusion  was  reached  in  Art.  127  that 


2/2  THE  ADJUSTMENT   OF   OBSERVATIONS. 

any  one  of  the  resulting  forms  of  side  equation  was  as 
good  as  any  other  in  satisfying  the  conditions  imposed. 
But  when  a  side  equation  is  reduced  to  the  linear  form 
and  is  no  longer  rigorous  the  question  deserves  farther 
notice. 

Two  points  are  to  be  considered — precision  of  results  and 
ease  of  computation.  As  regards  the  first,  since  the  differ- 
ences in  a  table  of  log.  sines  are  more  sharply  defined  for 
small  angles,  and  these  differences  are  the  coefficients  of 
the  unknowns  in  the  side  equation,  it  follows  that  in  general 
that  vertex  should  be  chosen  which  allows  the  introduction 
of  the  acutest  angles  into  the  side  equation. 

Labor  of  computation  will  be  saved  by  choosing  the  pole 
so  that  as  few  sine  terms  as  possible  enter.  Thus  by  choos- 
ing the  pole  at  O,  the  intersection  of  the  diagonals  (Fig.  22), 
the  side  equation  would  contain  8  terms,  whereas  if  taken 
at  any  of  the  vertices  only  6  terms  would  enter.  Also, 
other  things  being  equal,  we  should  choose  that  pole  which 
introduces  the  smallest  number  of  unknowns  into  the  equa- 
tion, for  then  the  normal  equations  would  be  more  easily 
formed. 

If  the  approximate  form  of  solution  in  Art.  115  is  em- 
ployed it  is  advantageous  to  choose  the  pole  at  the  inter- 
section, O,  of  the  diagonals,  as  will  be  seen  in  the  sequel. 

133.  Number  of  Side  Equations  in  a  Net. — A  line  being 
taken  as  a  base,  its  extremities  are  known.  To  fix  a  third 
point  we  must  know  the  other  two  sides  of  the  triangle  of 
which  this  point  is  to  be  the  vertex.  Hence  if  we  have  a 
net  of  triangles  connecting  s  stations,  two  of  the  stations 
being  the  ends  of  the  base,  we  must  have,  in  order  to  plot 
the  figure,  2(5  —  2)  lines  besides  the  base  ;  that  is,  2^—3 
lines  in  all. 

Starting  from  the  base,  each  line  in  this  figure  can  be 
computed  in  but  one  way,  but  any  additional  line,  whether 
observed  over  in  one  or  both  directions,  can  be  computed 
in  two  ways,  and  therefore  gives  rise  to  a  side  equation. 
If,  then,  the  total  number  of  lines  in  the  figure  is  /,  the  num- 


APPLICATION   TO   TRIANGULATION.  273 

her  of  side  equations,  as  indicated  by  the  number  of  super- 
fluous lines,  is 

/-2.J+3. 

134.  Check  of  the  Total  Number  of  Conditions.  —  Leaving 
local  equations  out  of  account,  if/  is  the  number  of  lines  in 
a  figure  sighted  over  in  both  directions,  and  s  the  number 
of  stations,  the  total  number  of  angles  in  the  figure  is  2/  —  s. 
If  4  of  these  lines  have  been  sighted  over  in  one  direction 
only,  the  number  of  angles  is  reduced  to  2/  —  /2  —  s. 

Now,  the  number  of  angle  equations  in  the  figure  is 


and  the  number  of  side  equations  is 

/-2J+3 

.*.  the  total  number  of  condition  equations  is 

2/  —  /2  —  35  -f-  4,      that  is,  11  —  2s  -{-  4 

where  ;/  is  the  number  of  angles  in  the  figure. 

But  we  have  seen  in  Art.  123  that  the  total  number  of 
conditions  to  be  satisfied  among  the  same  n  angles  is 

n  —  2s  -f-  4 

We  conclude,  therefore,  that  the  conditions  are  completely 
covered  by  the  angle  and  side  equations. 

135.  Manner  of  Selecting  the  Angle  and  Side 
Equations.  —  In  the  selection  of  the  angle  and  side  equa- 
tions in  a  triangulation  net  we  have  two  dangers  to  guard 
against:  first,  that  we  omit  no  necessary  conditions,  and, 
second,  that  we  admit  no  unnecessary  ones.  The  rule 
usually  followed  is  to  start  from  some  line  as  base,  and  plot, 
the  figure  proceeding  from  station  to  station, 
writing  down  the  conditions  that  express 
the  connections  of  each  station  to  the  net  as 
the  net  grows. 

For  example,  let  Fig.  27  represent  a  tri-        Fis-26 
angulation  net.     Taking  AB  as  base  (Fig.  26),  a  third  station, 
C,  gives   an    angle  equation  from   the    triangle   ABC,      A 


274 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


fourth  station,  D,  gives  in  addition  angle    equations  from 
ABD,ACD;  side  equation  from  ABCD.     A  fifth  point,  £, 
gives  in  addition  (Fig.  27)  angle  equations 
from    ABE,    ADE  ;    side    equation    from 
ABDE. 

We  have  thus  in  the  complete  figure 
(Fig.  27)  5  angle  equations  and  2  side  equa- 
tions. 


Fig.27 


Check.     Number  of  stations  =  5 
Number  of  lines       =  9 
.'.  Number  of  angle  equations 
and  Number  of  side  equations 

As  an  illustration  of  the  dif- 
ficulties which  may  arise  in  se- 
lecting the  angle  and  side  equa- 
tions, let  us  take  the  triangulation 
around  the  Chicago  Base  (1877). 
It  is  represented  in  the  figure. 

From  the  rules  laid  down  in 
Arts.  125,  133  it  follows  that  there 
are  in  the  adjustment  10  angle 
equations  and  8  side  equations. 

The  peculiarity  of  the  system 
is  that  the  station  F  is  very  close 
to  the  base  line  DE.  Thus  the 
angle  equation  from  the  triangle 
DEF 

EDF—  00°  oo'  oo".8i5+^ 
FED  =  00°  oo'  i". 1 854-1;, 
DFE=i79°  59'  56".733  +  *,  + 


=9  —  2x5+3  = 


Fig.28 


179°  59'  5* 

1 80°  oo'  oo".ooo 

o  =    -  i".267  4-  vt  4-  va  4-  v^  4-  v, 

In  the  selection  of  the  side  equations  it  is  advisable  to 
avoid  those  quadrilaterals  in  the  figure  which  are  entangled 


APPLICATION  TO   TRIANGULATION.  275 

with  the  above  triangle;  that  is,  the  quadrilaterals  A£DF9 
BDFE,  GDFE.  For  example,  if  we  take  the  quadrilateral 
GDFE  we  have,  in  units  of  the  seventh  place  of  decimals, 
pole  at  G,  the  side  equation 

54.772  i?,  —  o.oooSv,  -f-  o.ooi  5^4  —  4o.7o66(?a  -f~  ^«) 

-f-  26.  1  803*',  —  0.0003  (v»  +  *0  =  40.  1  3 

Now,  since  the  coefficients  of  ?„,  v4,  ?„,  ?„  are  each  less  than 
2  in  the  third  place,  this  equation  is  nearly  the  same  as 


54.772^  -  40.707(X  +  v.)  +  26.  iSo?7  =  40.  1  3 
or  dividing  by  40  and  replacing  ?&  -j-  ?0  by  —  v3  —  vt, 
1-369*'!  +  i.oi8(?s  +  v<)  +  0.654^  =  i 


which   is  nearly  the  same  as  the  angle  equation  from  the 
triangle  DEF. 

Similarly  the  quadrilaterals  AEDF,  BDEF  give  respec- 
tively, neglecting  coefficients  less  than  5  in  the  fifth  place, 


1.2192;,  +  o.2o8f7  +  0.722(2/3  -f-  ^4)  =  0.790 
0.256?,  +  2.466?,  +  1.3430s  +  O  =  0.836 

both  of  which  express  approximately  the  same  relations 
among  the  angles  of  the  small  triangle  DEF  as  the  angle 
equation  formed  from  this  triangle.  We  therefore  conclude 
that  in  the  formation  of  the  condition  equations  other 
quadrilaterals  than  these  should  be  chosen. 

136.  Again,  in  selecting  the  side  equations  in  a  net  care 
must  be  taken  that  only  independent  conditions  are  chosen. 
Thus  in  Fig.  28  we  might  have  chosen  the  following  eight 
quadrilaterals  from  which  to  form  the  side  equations: 

AGBD,  AGDE,  AGBE,   BDFG, 
BFEG,   AGDF,    ACBE,    BDEG. 

A  careful  examination  will  show  that  these  figures  are  not 
independent.  For,  taking  the  four,  AGBD,  AGDE,  AGBE, 

36 


276 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


BDEG,  and  choosing  G  as  pole,  which  is  common  to  all,  we 
have  the  side  equations 

sin  ADG  sin  DBG  sin  BAG  _  i 

~sm  DAG  sirHWG  sin  ABG 

sin  DAG  sin  EDG  sin  AEG 


sin  ADG  sin  DEG  sin  -ZL46P 
sin  y^C  sin  ^^^  sin 


—  i 


sn 
sin 


sin  EBG  sin  AEG 


sn 


sn 


sin  DBG  sin  EDG  sin  BEG 
which  equations  multiplied  together  reduce  to  the  identical 
form 

i  =  i 

showing  that  from  any  three  the  fourth  may  be  found. 

The  entrance  of  mutually  dependent  conditions  would, 
however,  be  detected  in  the  course  of  the  solution  of  the 
normal  equations,  as  we  should  arrive  at  two  identical  equa- 
tions ;  or,  in  other  words,  one  of  the  correlates  would  be- 
come indeterminate. 

If  the  rule  given  on  p.  273  is  followed  closely  this 
repetition  of  conditions  will  hardly  occur. 


Fig.30 


Fig. 32 


Fig.33 


Fig.  34 


APPLICATION   TO   TRIANGULATION. 


277 


137.  In  the  example  chosen  (Fig.  28)  the  selection  of  the 
angle  and  side  equations  may  be  made  in  the  following 
order,  proceeding  from  the  line  AB,  and  adding  station  to 
station  and  line  to  line  till  the  complete  figure  is  reached : 


From  Fig.  29, 

From  Fig.  30  in  addition, 

From  Fig.  31  in  addition, 
From  Fig.  32  in  addition, 

From  Fig.  33  in  addition, 
From  Fig.  34  in  addition, 


Angle  equations. 

Triang.  AGB 
AEB 
BEG 


DAG 
BDG 

DAE 

DAF 

BDF 

BFE 

BAG 


Side  equations. 

Quad.AGSE 

"  AGBD 

"  AGDE 

"  AGDF 

"  BDFG 

"  KFEG 

"       ACBE 
"       CBDE 


In  all  10  angle  equations  and  8  side  equations,  as  should 
be  (p.  274). 

138.  We  finally  notice  the  arrangement  for  solution  of  the 
condition  equations  in  the  net  adjustment.  On  first  thoughts 
it  might  seem  that  it  would  be  well  to  arrange  the  angle 
equations  and  side  equations  in  two  separate  sets,  and  so 
carry  them  forward  for  solution.  This  was  done  in  some  of 
the  older  work.*  The  only  objection  to  doing  it  is  that  the 
process  of  finding  the  corrections  is  more  troublesome.  Ex- 

*  See  for  an  example  Verification  and  Extension  of  La  Cailles  Arc  of ' Meridian,  by  Sir 
Thomas  Maclear,  vol.  i.  pp.  496  seq. 


278 


THE   ADJUSTMENT   OF    OBSERVATIONS. 


perience  shows  that  the  solution  of  a  series  of  normal  equa- 
tions is  much  facilitated  if  the  coefficients  are  arranged  as 
the  steps  of  a  stair  rather  than  irregularly.  Thus 

\ad\x-\-\aU\y  =  [al~\ 

\ab~\x  +  \bU\y  -f  \bc~\s  =  \bl] 

\bc\y +\cc-\z=\cl} 

is  a  more  convenient  form  for  solution  than 

\aa~\x  -f-  [ab]  y  =  [«/] 


Fig.  35 


The  condition  equations  should  therefore  be  so  arranged 
that,  as  far  as  possible  (and  it  cannot  always  be  done  at  the 
first  trial),  the  normal  equations  will  fall  in  the  first  of  the 
above  forms  rather  than  the  second.  A  good  rule  is  to 
begin  with  an  angle  equation,  proceeding  from  triangle  to 
triangle  until  the  points  gone  over  are  covered  by  a  side 
equation,  and  then  introduce  it.  Continue  the  process  with 
the  remaining  sets  of  triangles. 

As  to   the   angle   equations   themselves,  it  matters  but 
little  which  triangles  are  taken  to  form  them.     It  is  better, 

however,  to  avoid  triangles  with 
very  small  angles,  such  as  DEF 
in  Fig.  28,  and,  where  angles  so 
very  small  occur,  to  avoid  tri- 
angles involving  angles  imme- 
diately contiguous  to  these  small 
angles. 

139.  In  the  explanation  of  the 
formation  of  the  side  equations 
we  have  assumed  that,  the  com- 
putation of  one  side  from  another 
assumed  as  base,  by  different 
routes  we  proceed  through  chains  of  triangles.  The 
omission  of  certain  lines  in  the  measurement  may  make  it 
necessary  to  proceed  through  polygons,  and  then  the  for- 
mation of  the  side  equations  becomes  more  complicated. 


APPLICATION   TO   TRIANGULATION.  279 

A  good  illustration  occurs  in  the  triangulation  of  Lake 
Superior*  (1871)  near  Keweenaw  Base,  as  shown  in  Fig.  35. 

Here  there  are  18  lines  and  8  stations,  requiring  5  side 
equations.  Proceeding  from  the  line  DG  and  adding  point 
to  point,  we  form  the  first  four  side  equations  from  the 
quadrilaterals 

HGDE,  HGEF,  HEFC,  CEFB 

With  these  there  is  no  difficulty. 

The  fifth  side  equation  is  furnished  by  the  pentagon 
ABCED.  We  cannot  compute  directly  any  specified  side 
of  this  pentagon  from  another  side  through  triangles  only. 
A  little  artifice,  however,  will  enable  us  to  do  this.  Sup- 
pose the  line  CD  drawn.  Call  the  angle  GDC=,r  and 
ECD=}>.  This  new  line  CD  gives  an  additional  side 
equation.  We  take  the  two  pentagons  ABCED,  DEGFC. 

From  the  first  pentagon,  ABCED, 

CE     CD     CA     CB__ 
~CD     CA     CB     CE~ 
or 

sin  (EDG  -f-  -*")  sin  CAD         sin  CBA    sin  CEB  _ 

sin  CED         sin  (ADG  -\-  x)  sm  CAB  sin  CBE  = 

and  from  the  second  pentagon,  DEGFC,  pole  at  E, 

EG    EF    EC 


^ 
EG    EF    EC    ED 

or 

sin  EGD  sin  EFG    sin  ECF  sin  (EDG-\-x]  _ 
sin  EDG  sin  EGF  sin  EFC  sin,)' 

Now,j  can  be  expressed  in  terms  of  x  from  the  triangle 
CDE;  thus 


(3) 
where  f  is  the  spherical  excess  of  the  triangle  CDE. 

*  Report  of  Chief  of  Engineers,  1872. 


280  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Eliminating  x  from  equations  i,  2,  3,  and  the  required 
side  equation  results. 

To  find  it  write  (i)  in  the  form 

cot  (EDO  -f  x) 

i  sin  CAD    sin  ABC    sin  EEC 

T-,  —  cot  ADE 


sin  ADE    sin  CED    sin  BAC    sin 

from  which  the  value  of  x  follows  at  once. 

Hence  since  y  is  known  from  (3),  all  the  angles  in  (2) 
are  known,  and  the  solution  is  finished  in  the  usual  way. 

If  chains  of  triangles  intersect  so  as  to  form  a  closed 
polygon,  this  method  may  be  employed  in  simple  cases. 
Such  forms  are,  however,  in  general  better  treated  by 
other  methods,  which  will  be  found  in  works  on  the  higher 
geodesy. 


140.  Ex.—  Adjustment  of  the  Quadrilateral  WSOLi  (Fig.  19). 

The  method  of  forming  the  condition  equations  having  now  been  ex- 
plained, we  are  ready  to  adjust  the  quadrilateral  NSOL,  as  promised  in 
Art.  120. 

The  condition  equations  have  all  been  formed  in  the  preceding  sections. 
Collecting  them,  we  have  : 

Local  equations  (Ex.  i,  2,  Art.  121) 

•v\  +  v?.  +  z>3  =  —  1-37 
w4  4-  v*,  —  v6  =  —  1.07 

Angle  equations  (Ex.  Art.  124) 

Vs  +  V4  +  V^  —  —  0.48 

V6  +  VT  +  VS  +  Vu  =  —  1.  10 

Side  equation,  the  unit  being  the  sixth  place  of  decimals  (Ex.  Art.  130), 
1.43^1  —  0.92^2  —  i.  93^6  +  0.747/0  —  0.43^7  +  i.>7z/8  =  —  2.00 

The  methods  of  solution  have  been  explained  in  Chap.  V.,  and  we  shall  pro- 
ceed in  the  order  there  given  for  the  four  forms. 


APPLICATION   TO   TRIANGULATION. 


281 


FIRST  SOLUTION — METHOD  OF  INDEPENDENT  UNKNOWNS. 

There  being  9  unknowns  and  5  condition  equations  connecting  them, 
there  must  be  4  independent  unknowns.  We  shall  choose  vt,  7^,  vt,  7/0. 
Expressing  all  of  the  unknowns  in  terms  of  these  four,  we  write  the  equations 
in  the  form  of  observation  equations,  as  follows  (see  Art.  109): 


7'3=  + 

Va  =  -  Vi  — 

7'4  — 


7-B  =    —  0.5657',    + 
7-3  =   —  0.435T-1   — 


-i-37 


i- 


7/4 


weight  2 
"  2 
"  14 
"  23 

6 


+  7-4   +  7-o   +    1.07 

7-2  —  7'4  +   0.89 

37'2  —  0.6617-4    +   0.672775  —   I.36l 
37/2    +  O.66l7'4   —   r.6727'5  —   1.699 


Hence  the  normal  equations 


7-1 

»« 

»4 

V& 

Const. 

+  48.83 

+  50.70 

-    32-93 

+    5-44 

-  53-45 

+  72.45 

-    40.33 

+  24.09 

-  69.70 

+    64.93 

-    2.29 

+  28.18 

+  35-82 

—  29.30 

+  83.79 

=  w 

Solving  these  equations    (page  284),    we    have    the   values   of  the   cor- 
rections 

Vi  =    —  O".S2  7/4=   —  O".22 

•v-i  =  — o".36  5-0=  — o"_47 

and  thence  from  the  condition  equations 

7^3=  —  o".ig  Z'K—  —  i".33 

z/o  =  +0^.38  v.j=  —  o".oS 

z/7  =  —  o".o7 

These  corrections  applied  to  the  measured  values  of  the  angles  give  the  most 
probable  values  as  follows  : 


Ml  -  124°  09'  39". 87 

^/2  =  ii3°  39'  04". 71 

-d/3  =  i22°  n'  15".  42 

Jlft=    23°  08'  05  ".04 

M,,—    47°  31'  19". 94 


Ma  =  70°  39'  24". 98 

-rt/7=34°  40'  39"- 59 

^/e  =  43°  46'  25". 07 

Mv  —  ^o  53'  30".  73 


282  THE  ADJUSTMENT   OF   OBSERVATIONS. 

The  Precision  of  the  Adjusted  Values.  —  (a)  To  find  the  m.  s.  e.  of  an  obser- 
vation of  the  unit  of  weight  (Arts.  99,  101). 
From  the  above  values  of  the  residuals  v 

[pw]  =  7-  53 

Check   of  [/TO].     Carrying  through  the  solution  of  the  normal  equations  the 
extra  column  required  by  the  sum  [///],  we  find  (page  284) 


Hence 


9-4 
=  ±  i".23 

(b)  To  find  the  weight  and  m.  s.  e.  of  the  adjusted  value  of  an  angle. 
Take  the  angle  NLS.     Proceeding  as  in  Art.  101,  we  have 

F=NLS 

=  180  +  £  —  (M-i  +  v*  +  Mt,  +  v6) 

.'.    C/f=  —  772  —  7^6 

Hence  from   the  extra  column,  the  sixth,  carried  through  the  solution  of  the 
normal  equations  (page  284), 

UF  =  0.053 
and  therefore 


Up-  1.23  1/0.053 


(c)  To  find  the  weight  and  m.  s.  e.  of  the  adjusted  value  of  a  side,  the 
base,  NS,  being  supposed  to  be  free  from  error. 
Let  us  take  the  side  OL.     We  have 

F=  OL 

_         S'JL^^  sin  Lso 

~  sin  OLS 


sin  {Mi  +  VT)  sin  (M»  +  v») 
For  check  we  shall  proceed  in  two  ways. 


APPLICATION   TO   TRIANGULATION.  283 

(i)  Expand  /'directly;  then 

/  6F  6F  6F  6F       \ 

dF=  (  TTT  *'»  +  nrr  v«  +  TUT  ''•>  +  ^nrr  T'» )  sin  l 
\dMs  dM*  ^M^  dMa      ) 

—  0.05057/3  +  0.02827'e  —  O.Il607-7  —  0.13427-9 

=  —0.0077/1    +0.1717-2    +  0.0567/4    +0.2537/5 

by  substituting  for  7-3,  va,  ?'-,  ''a  their  values  from  equations  i. 

Carry  through  the  solution  of  the  normal  equations  the  extra  column  re- 
quired by  these  coefficients,  and  (see  page  284) 

UF  =  0.0019 
Hence 


I.IF  =1.23  V 0.0019 
=  o'".o5 

(2)  Take  logs,  of  both  members  of  the  equation  ;  then 

log  /'—log  NS  +  log  sin  (M3  +  v3)  +  log  sin  (M»  +  z>6) 

—  log  sin  (A/,  +  v-,)  —  log  sin  (M*  +  v 

But  since  JVS  is  constant,  we  have,  in  units  of  the  sixth  place  of  decimals, 

f/log  F=  —  1.337-3  +  0.747-8  —  3-04^7  —  3-527'a 

=  —0.187-1  +  4.5O7'2  +  1.457/4  4-  6.637-5  from  equations  r. 

Hence  from  the  last  column  addejd  to  the  solution  of  the  normal  equations, 

u ,o  F=  1.50  in  units  of  the  sixth  place  of  decimals. 
Also, 

^togF=I-23  V^° 

=  1.5  in  units  of  the  sixth  place  of  decimals. 
Now,  since  (p.  23) 

dF 

(/log  F=  —  mod 
t1 

and  (p.  255) 

F=  16556^, 
.-.  f.ip  —    om.o6. 

37 


284 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


The  solution  of  the  normal  equations,  with  the  extra  columns  required  by 
the  weight  determinations,  is  as  follows  : 


-k 

V, 

*4 

* 

I 

/(angle). 

/(side). 

/(side). 

+  48.83 

+  50.70 
+  72.45 

-  32.Q3 
-  40.83 
+  04.93 

+    5-44 
+  24.09 
-    2.29 
+  35.82 
[///]  = 

-  53-45 
-  69.70 
+  28.18 
-  29.30 
+  83.79 

-  i. 

-  0.007 
+  0.171 
+  0.056 
+  0.253 

-  0.18 
+  4-50 
+  1-45 
+  6.03 

+  I. 

+   1.0383 

-   0.6743 

+    0.1114 

-    1.0946 

-  O.OOOI 

-  0.0037 

+  19.8082 

-     6.6430 
+  42.7253 

+  18.4420 
+     1-3784 
+  35.2140 

+  14.2038 
+    7.8652 
+  23.3454 
+  25.2836 

-  i. 
o. 

+  0.1761 
+  0.0527 

+  0.2535 

0. 

+  4.6871 
+  1.3285 
+  6.6501 
+  0.0007 

+  I. 

-    0.3354 

+    0.9310 

+    0.7176 

+  0.0505 

+  0.0089 

+  0.2366 

+  40.4972 

+    7.5630 
+  18.0445 

+  12.6322 
+  10.1114 
+  15.0910 

+  0.0505 

+  o.uiS 
+  0.0894 

+   O.OOIO 

+  2.9002 
+  2.2867 
+  1.1089 

+  I. 

+    0.1868 

+    0.3119 

-  0.0083 

+  0.0027 

+  0.0716 

+  16.6317 

+  11.1516 

-  0.0057 
+  0.0028 

+  0.0690 
+  0.0003 

+  1.7452 
+  0.2076 

+  i. 

+    0.4661 

-  0.0004 

+  0.0003 

+  0.1049 

[pvv\  = 

+    7-5376 

o. 

o.oooo 
0.0505 
0.0028 
o.oooo 

o. 

o.oooo 
0.0016 
0.0003 
o.oooo 

+  0.1832 

0.0007 
1.1089 
0.2076 
0.1832 

0.0533 

Jf 

1.5004 

The  solution  has  been  carried  through  to  four  places  of  decimals,  on  account 
of  loss  of  accuracy  arising  from  dropping  figures  in  multiplications.  The  re- 
sulting values  of  the  corrections  have  been  cut  down  to  two  places  of  deci- 
mals. The  work  was  done  with  a  machine,  as  explained  on  p.  161,  the  recip- 
rocals of  the  diagonal  terms  being  used  so  as  to  avoid  divisions.  Thus  the 
first  reciprocal  is  0.02048. 

SECOND  SOLUTION — METHOD  OF  CORRELATES. 
Arranging  the  condition  equations  in  tabular  form,  we  have 


weights    2 

2 

J4 

23 

6 

7 

31 

I 

8 

i    1-43 

-  0.92 

-  i-93 

+  0.74 

-  0.43 

+  1.77 

-  2.00 

(-  i. 

+    I. 

+    I. 

-  1.37 

+    I. 

+    I. 

+  I. 

-  0.48 

+    I. 

+  i. 

-  i. 

-  1.07 

+  i. 

+  I. 

+  I. 

+  i. 

-    1.  10 

APPLICATION   TO   TRIANGULATION. 

The  Correlate  Equations. 


232/4  = 


I. 

II. 

III. 

IV. 

V. 

+  1-43 

+  i. 

• 

-0.92 

+  i. 

+  i. 

+  i. 

+  i. 

+  I. 

-i-93 

+  I. 

+  0.74 

—  I. 

+  I. 

-0.43 

4-  i. 

+  I. 

+  1-77 

+  I. 

+  I. 

The  Normal   Equations. 


I. 

II. 

III. 

-     IV. 

V. 

/ 

+  5.284 

+  0.255 

—  0.014 

-0.427 

+  1.862 

—  2.OO 

+  1.071 

+  0.071 

-1-37 

+  0.147 

+  0.043 

+  0.032 

—  0.48 

+  0.353 

-0.143 

-1.07 

+  1.300 

—  I.IO 

The  solution  of  these  equations  gives  (see  page  287) 

I.  =  —0.3973 
II.  =  -  1.0749 

III.  =  -  i.  6006 

IV.  =  -3.5/21 
V.  =  —  0.6301 

Substituting  these  values   in    the  correlate  equations,  the  same  values  of  the 
corrections  result  as  before.     Also, 


=  7-  53 


286  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  Precision.  —  (a)  To  find  the  m.  s.  e.  /<  of  an  observation  of  weight  unity. 
From  the  values  of  v  we  find  directly 

[pvv]  =  7-53 

Checks  of  [pf'v~\.  These  are  worked  out  in  the  sol  ution  of  the  normal  equa- 
tions on  page  287,  according  to  the  formulas  of  Art.  in,  and  give  7.54  and  7.55 
respectively. 

Hence  taking  the  mean,  [/w]  =  7-54,  and  the  number  of  conditions 
being  5, 


=  i  ".23  as  before. 

Compare  Ex.  2,  Art.  in. 

(b)  To  find  the  weight  and  m.  s.  e.  of  the  adjusted  value  of  an  angie. 
Take  the  angle  NLS. 


From  the  values  of  u,  a,  !>,...  in  the  condition  equations  in  connection  with 
the  values  of  f  given  by  this  function,  we  have 

[naf]  =  +  0.782  [«<(/"]  =  —  0.167 

\ubf~\  —  —  0.500  [U£/]  =       o. 

[ucf]  =       o.  [uff]  =  +0.667 

Hence   from    the   seventh  column   in  the  solution  of  the  normal  equations 
(page  287), 

uF  =  0.053 
and 


fip=  1.23     o.o53 

=  0".2S 

Compare  Ex.  4,  Art.  in. 

(c)  To  find   the  weight  and  mean-square  error  of  the  adjusted  value  of  a 
side,  the  base  being  free  from  error. 

Take  the  side  Oneota-Lester. 

As  in  (c),  page  282,  we  have 

dF~=  —0.05057/3  +  0.02822/6  —  0.1160277  —0.13422/9 
Also  from  the  condition  equations 

[uaf]  =  +  0.0046  [«"/]  =  —  0.0040 

[«£/]  =  —  0.0036  [«*/]  =  —  0.0165 

[ucf]  =  —  0.0073  [«//]  =  +  0.0030 


APPLICATION   TO    TRIANGULATION.  287 

Hence  from  the  eighth  column  in  the  solution  of  the  normal  equations, 

UP  —  0.0023 
and  finally, 


HP  —  1.23  V.OO23 
=  o"».o6 

Solution   of  the  Normal    Equations. 


I. 

ii. 

III. 

IV. 

V. 

I 

/(angle). 

/(side). 

4    5-284 

+  1.071 

4   O.C7I 
+  0.147 

+  0.043 
+  0.353 

4   O.O32 
-   0.143 
4-   I.30O 

-  1-37 
-  0.48 
-  1.07 

-    I.  10 

-  0.500 
-  0.167 

4    0.667 

-  0.0036 
-  0.0073 
-  0.0040 
-  0.0165 

4   0.0030 

+    I. 

4   0.0483 
4    1.0587 

-  O.OO26 

4  0.0717 
4   0.1470 

-  0.0808 

4   O.O2O6 
4  0.0419 
4    0.3185 

4-  0.3524 

-  O.OQ02 
4    0.0369 

4    0.0075 
4-  0.6438 

-  0.3785 

-  0.4853 
-  1.2316 
-  0.3952 
+  0.7570 

4  0.1480 

-  0.5377 
4  O.002I 
-  0.1035 
-  0.2756 
4  O.55IO 

+   O.OOO9 

-   0.0038 
-   0.0073 
-  0.0036 
-  0.0182 

4    I. 

4  0.0677 
4   O.I42I 

4   0.0204 

4    0.0404 
4   0.3l8t 

-  0.0851 

4  0.0434 
4  O.O093 
+  0.6361 

-  1.2025 

-  0.3991 
-  1.2068 
-  0.5037 

4    I  .  5309 

-  o  5078 

4  0.0385 
-  0.0930 
-  0.3214 
4   0.2780 

-  o  0036 

-  O.OO/O 
-  0.0035 
-  0.0185 
4  O.O030 

Values  of  the  Unknowns 

I.  =  -  0.39 
II.  =  -  1.07 

m.=  -  1.60. 

ll~  =  ~  3'E 
\  .  =  -  0.631 

4    1. 

4   0.2843 
4   0.3066 

+   0.3054 

-  0.0030 
+  0.6228 

-   2.8o86 
-    1.0933 

-  o  3818 

4    I.I209 

4-   0.2709 

-  0.1039 
-  0.3332 
4   0.2676 

-  0.0493 

-  O.OOI5 
-  0.0164 

+  o  0027 

73 

% 
il 

31 

4    I. 

-  0.0098 
4  0.6228 

-   3.5659 

-   0.^924 
4   3.8986 

-   0.3389 

-  0.3342 
+  0.2324 

-  O.0049 

-  0.0164 
4    0.0027 

4   I. 

-  o  6301 

+   0.2472 

-  0.5366 

4  0.0531 

-   0.0264 
4-   0.0023 

I.       1'        =  -  o  3973  x    -  2.00  =  0.79 
II.       /"            -  1.0749  x    -  1.37=  1.47 
III.       1'"     =  -  1.6006  x    -0.48  =  0.77 
IV.        /""           -  3.^721    x    _   1.07  =  3.82 
V.       /"'"=-  0.6301  x    -  1.10  =  0.69 

0.7^70 
1.5309 
1  .12Oq 
3.8986 
0.2472 

7-54 

7.5546 

The  values  of  [/>fzp]  are  found  from  equations  2,  3,  Art.  in. 


288  THE   ADJUSTMENT   OF   OBSERVATIONS. 

THIRD  SOLUTION — SOLUTION  IN  Two  GROUPS. 
The  form  given  in  Art.  113  is  followed. 

The  Local  Adjustment. 

(a)  At  North  Base. 
The  Observation  Equations. 


p 

c*o 

(*a) 

/ 

2 

+  I 

O. 

2 

+  I 

0. 

14 

—  I 

—  I 

-1-37 

The  Normal   Equations. 

(*i)    (**) 

16  +  14  —  —  19.18  =  [pal]     suppose 
14  +  16—  —  19.18  = 


Solving  in  general   terms. 

(Xl)  =  +  0.267  [pal]  —  0.233 


Hence 


(jr2)  =  -  0.233  [pal]  +  0.267  [pbl] 

(xt)  =  —  o".64 
(jT2)  =  —  o".64 

(x3)  =  +  o".64  +  0^.64  —  i".37 
=  —  o".og 


and 


Local  Angles. 
124°     09'     40". 05 
113°     39'     04". 43 

122°       II'       15".  52 

To  find  the  m.  s.  e.  of  a  single  observation. 
The  value  of  [pvv]  =  [pxx]  =  1.75. 
Hence  for  this  station,  the  number  of  conditions  being  3—2, 


=1/I« 

r    o  — 


=  i  -3 


APPLICATION   TO    TRIANGULATION. 

(b)  At  South  Base. 
The  Observation   Equations. 


/ 

(*0 

(*») 

/ 

23 

+  I 

o. 

6 

+.1 

o. 

7 

+  I 

+  I 

-1.07 

The  Normal   Equations. 


289 


Hence 


Also, 


30  +     7  =  -  7.49 
7  +  13  =  -7-49 


(jr6)  =  —  o".so 

(.*•«)  =  —  o".i3  —  o".5O  +  i  ".07 
=  +  o"-44 

Local   Angles. 

23°     08'     05".  13 

47°     3i'     IQ"^1 
70°     39'     25".  04 


=  3.24 


The  General  Adjustment. 
Most   Probable  Angles. 

At  N.  Base,  124"  09'  40". 05  +  (i) 
113°  39'  04". 43 +  (2) 
122°  n'  15".  52  —  (i)  —  (2) 

At  S.  Base,         23°     08'     05".  13  +  (4) 

47°     3i'     i9"-<)i  +  (5) 

70°     39'     25".04  +  (4)  +  (5) 

At  Oneota,          34°     40'     39". 66  +  (7) 
43°     46'     26".40  +  (8) 

At  Lester,  30°     53'     30". Si  +  (9) 


290  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  Angle  and  Side  Equations. 

t 

(a)  Triangle,  N.  Base,  S.  Base,  Oneota. 

Angle  SNO  122°  n'  15". 52  —  (i)  —  (2) 
"       NSO    23°  08'  05".  13 +  (4) 
"       NOS    34°  40'  39". 66 +  (7) 
Sum        =  180°  oo'  oo".3i 
180  +•  £  —  180°  oo'  oo'.os 

o=o".26-(i)-(2)  +  (4)  +  (7) 

(b)  Triangle  Lester,  Oneota,  S.  Base. 

Angle  NSO  70°  39'  25". 04  +  (4)  +  (5) 
SOL  78°  27'  06". 06  +  (7)  +  (8) 
OLS  30°  53'  30". 81  +  (9) 

180°  oo'  oi".9i 
180°  oo'  oo".37 

0=1  ".54  +  (4)  +  (5)  +  (7)  +  (8)  +  (9) 


(c)  Quadrilateral  N.  Base,  S.  Base,  Oneota,  Lester. 

sin  LNS   sinLSO  sin  LOW  _ 

sin  LNO  sin  NSL  sin  LOS  ~ 

LNS  =  113°  39'  04". 43  +  (2)  LNO  —  124°  09'  40". 05  +  (i) 

LSO  —    70°  39'  25". 04  +  (4)  +  (5)  NSL  =    47°  31'  19". 91  +  (5) 

LON=    43°  46'  26". 40  +  (8)  LOS  =    78°  27'  06". 06  +  (7)  +  (8) 

9.9618975,6—    9,22    (2)  9.9177479,3  —  14,29(1) 

9.9747660,1+    7,391(4)  +  (S)}  9.8677849,8  +  19,28(5) 

9.8399903,4  +  21,98(8)  9.9911180,3+    4,3o|(7)  +  (8)[ 

539.1  509.4 

509,4 

29,7 


APPLICATION  TO   TRIANGULATION.                        29! 

Check  by  deducting  J-  of  the  spherical  excesses  of  the  triangles  from  the 
angles. 

113°  39'  04". 36  124°  09'  40". 01 

70°  39'  24". 92  47°  31'  19". 84 

43°  46'  26". 36  78°  27'  05". 93 

9.9618976,2  9.9177479,9 

9-9747659.3  9.8677848,6 

9.8399902,5  9.9911179,8 

38,0  8,3 
8,3 


29,7 
The  two  methods  agree  well. 

A  glance  at  the  log.  differences  for  i"  shows  that  by  expressing  them  in 
units  of  the  sixth  place  of  decimals  their  average  value  is  unity  nearly.  We 
have,  then,  for  the  side  equation, 

i. 43(1) -0.92(2)  +  0.74(4)-  1. 19(5) -o.43(7)  +  1-77(8)  +  2.97  =  0 

The  Weight  Equations. 
(0  =  —0.233  |jj  +0.267  |~2~| 
(2)  =  +0.267  I  i  |  —0.233  I  2  | 

(4)  =  +0.038   |  4  |    —0.021   |  j>J 

(5)=  —0.021  j~4~j  +0.088  |~5~j 

(7)=  +0.032  |~7~| 

(8)=  +i.ooo|T] 

(9)=  +0.125  |  9  | 

The  Correlate  Equations. 


I. 

ii. 

in. 

Check. 

Qj=  -' 

+  1.43 

-  0-43 

|_2J    =      -  I 

—  0.92 

+    1.92 

|Tj=    +i 

+-  1 

+  0.74 

-  2.74 

ULJ  = 

+  1 

-  1.19 

+  0.19 

IT)  =    +  i 

+  1 

—0.43 

—  1-57 

|_8J  = 

+  1 

+  1.77 

-  2.77 

nn  = 

+  1 

—    1.  00 

The  check  is  formed  by  adding  each  horizontal  row  (Art.  84). 
38 


292 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


Expression   of  the  Corrections  in  Terms  of  the  Correlates. 


I. 

+  0.233 
—  0.267 

II. 

III. 
~  0.333 
—  0.246 

Check. 
4-  O.  IOO 

+   0.513 

-  0.034 

-  0.579 

+   0.613 

(2)  = 

—  0.267 
+  0.233 

+  0.382 
+   0.214 

—  O.II5 
-  0.447 

-  0.034 

+   0.596 

—  0.562 

(4)  = 

+  0.038 

+   0.038 
—  O.02I 

+   0.028 
+   O.O24 

—  O.IO4 
—  0.004 

+  0.038 

+    0.017 

+   0.052 

—  O.IO8 

(5)  = 

—  O.O2I 

—  O.02I 
+   0.088 

—  0.016 
—  0.105 

+  0.058 
+   0.017 

+    0.067 

—  O.I2I 

+  0.075 

(7)  = 

+    0.032 

+  O.O32 

—  0.014 

—  0.050 

(8)  = 

+    I. 

+  1.770 

-  2.770 

(9)  = 

+  0.125 

—  O.I25 

The  Corrections  in  Terms  of  the  Correlates  (collected}. 


(2)  = 

(4)  = 

(5)  = 

(7)  = 

(8)  = 

(9)  = 


I. 

-  0.034 

-  0.034 
+  0.038 

—  O.O2I 
+  0.032 


II. 


+  0.017 
+  0.067 
+  0.032 
+  I.OOO 

+  0.125 


III. 

-  0.579 
+  0.596 
+  0.052 

—  0. 121 

—  O.OI4 
+  1.770 


Formation  of  the  Normal   Equations. 


-(2) 

+  (4) 
+  (7) 


I. 

+  0.034 
+  0.034 
+  0.038 
+  0.032 

[+  0.138 


+  0.017 
+  0.032 

+  0.049 


III. 
+  0.579 

—  0.596 
4-  0.052 

—  0.014 


O.O2I 


Check. 

—  O.6I3 
+  0.562 

—  o.  108 

—  0.050 

+  o.  209 


1(4) 
(5) 
i<7) 
(8) 
(9) 


+  0.017 
+  0.067 
+  0.032 
+  I. 
+  0.125 


+  O.O52 

—  0.  121 

—  0.014 

+  1.770 


—  0.108 
+  0.075 

—  0.050 
-  2.770 

—  0.125 


+  0.049 


+  1.241 


1.687 


-  2.978 


APPLICATION   TO   TKIANGULATION. 


293 


II. 


-  0.92(2) 
+  0.74(4) 

-  i.i'X5) 

-  0.43(7) 
+  1-77(8) 


1.687 


III. 

+  0.852 
+  0.533 
-f-  0.038 
+  0.144 
+  0.006 
+  3-133 

+  4.706 


Check. 

—  0.804 

-  0.564 

—  o.oSo 

—  0.089 

+  O.O22 

-  4-9°3 

-  6.418 


The  Normal  Equations  (collected}. 

I.  II.  III. 

+  0.138  +  0.049  +  0.021  =  —  0.260 

+  1.241  +  1.687  =  — 1.540 

+  4.706=  —  2.970 

The  solution  of  these  equations  gives  (page  295) 

I.  =  -  1.597 

II.  =  -0.642 

III.  =  —  0.394 

Substitute  for  I.,  II.,  III.  their  values  in  (4),  and  we  have  the  general  cor- 
rections. 

Adding  the  local  corrections  and  general  corrections  together,  the  total 
corrections  to  the  measured  angles  result  and  are  as  follows  : 


Local. 

General. 

Total. 

p 

pvv 

Final  Angles. 

-fi  = 

—  o" 

64 

—  o" 

18 

=  —  o" 

82 

2 

i-34 

124° 

09' 

39".  §7 

-»'2   = 

—  o" 

64 

+  o" 

28 

=  —  o" 

36 

2 

.26 

H3° 

39' 

4"-7i 

JCa  = 

—  o" 

09 

—  o" 

10 

=  —  o" 

19 

14 

•50 

122° 

II 

i5"-42 

-T4   = 

—  o" 

13 

—  o" 

09 

=  —  o" 

22 

23 

I.IO 

23° 

08' 

5"-  04 

-V5  = 

—  o" 

50 

+  o" 

04 

=  —  o" 

46 

6 

1.27 

47° 

3i' 

!9"-95 

X*  = 

+  o" 

44 

—  o" 

05 

=  +  o" 

39 

7 

i.  06 

70° 

39' 

24  ".99 

X  7    = 

—  o" 

07 

=  —  o" 

o? 

3i 

•  15 

34J 

40' 

39"-  59 

Xf  = 

—  l" 

33 

—   —   I  " 

33 

i 

i-77 

43" 

46' 

25"-07 

-ra  = 

—  o" 

08 

=  —  o" 

08 

8 

•  05 

3«  ' 

53' 

30"-  73 

[/H 

=  7-50 

Number  of  local  conditions        =  2 
Number  of  general  conditions  =  3 

Total      =  5 

The  method  of  solution  just  given  is  substantially  the  same  as  that  em- 
ployed on  the  survey  of  the  Great  Lakes  between  Canada  and  the  United 
States  by  the  U.  S.  Engineers. 


294  THE  ADJUSTMENT   OF   OBSERVATIONS. 

The  Precision  of  the  Adjusted  Values. 

(a)  To  find  the  m.  s.  e.  of  an  observation  of  weight  unity. 

Computation  of  [/r^]. 

(r)  From  the  preceding  table  [/w]  has  been  found  directly  ;  thus 

[>»»]  =  7. 50 

(2)  Check  (Art.  114).     From  the  station  adjustments  find  [v°v°] 

N.  Base  gives  (p.  288)  1.75 
S.  Base  gives  (p.  289)  3.24 


4.99  =  [z/V]. 

From  the  general  adjustment  find  [ww]. 

(<r]  /„'  X  I.  =  -0.26  X  -  1-597=  +0.42 
/„"  X  II.  =  —  1-54  X  —0.642=  +  0.99 
/„"'  X  III.  = —2.97  X  —0.394=  +  1. 16 

+  2.57 

(/5)/0'      X  f===f=- 0.26  X -1.885=  +0.49 

L«Aj 

/o"l 

/„  .1  XF=z:  =  -  1.45  X  —  1.183=  +  1.72 
OB.I] 

/o'"2 

/„    .2  Xr--=^  =  —  °-94  X  —  0.394  =  +  0.37 
[fC.2] 

+  2.58 

.'.   [ww]         —  2.58 

and  [p-cn>]  =4.99+  2.58 

=  7-57 
Hence  taking  the  mean  of  the  values  of  \_pvv\, 


there  being  2  local  conditions  and  3  net  conditions. 

(b)  To  find  the  m.  s.  e.  of  an  angle  in  the  adjusted  figure. 
Angle  =  NLS 

...    ^=_(2)_(5) 

=  +  O.O55  I-  —  0.067  II-  +  O.7OO  III. 

from  the  weight  equations. 

From  equations  25,  Art.  114, 


APPLICATION   TO   TRIANGULATION.  295 

The  values  of  [anr],  [a/if]  .   .   .  are  given  in  the  weight  equations.     Hence 

q\  =+0.267  X  0  —  0.233  X  —  i  =+0.233 
q-i  =  —  o.  233  X  o  +  o.  267  X  —  I  =  —  o.  267 
q3  =  +0.038  X  o  —  0.021  X  —  I  =  +0.021 
q^  =  —  0.021  X  0  +  0.088  X  —  I  =  —  0.088 


g 

1 

gq 

o 

+  0.233 

o. 

—  I 

—  0.267 

0.267 

o 

+  O.O2I 

o. 

—  I 

—  0.088 

0.088 

l> 


=  O.O22. 


(See  the  solution  of  the 
normal  equations.) 


[VC.2] 


=  0.006 


=  0.274 


0.302 


=      °-355 
—  0.302 


uF  =  +  0.053 


.'.   /v  =  i".23  Vo.053 

=  0".28 

(c)  To  find  the  m.  s.  e.  of  a  side  in  the  adjusted  figure. 
Side  =  Oneota-Lesier. 

i"  OSL 


--- 

sin  SON  sin  OLS 

Therefore 

dF  =  1.33(1)  +  L33(2)  +  0.74(4)  +  o.74(5)  -  3-04(7)  -  3-52(9) 
in  units  of  the  sixth  place  of  decimals, 

—  0.174  I  -  —  0.475  II-  +  °-OI5  HI- 

from  the  weight  equations. 

The  solution  is  carried  through  exactly  as  in  the  preceding  case.    We  find 

[gq~\  —  2.01  r    and  UF  —  1.49 
Hence  H  VUF  =  1.23  V  1.49 

=  1.5  in  units  of  the  sixth  place  of  decimals. 

Now  log  0Z  =  4.2189699  OL-  16556"* 

16556 

.  '.  m.  s.  e.  of  side  =  —     -  X  0.0000015 
o.434 

=  ow.o6 


296  THE  ADJUSTMENT   OF   OBSERVATIONS. 

Solution  of  the  Normal   Equations. 


I. 

II. 

III. 

/ 

/(angle) 

+  0.138 

+  0.049 
+  1.241 

+  O.O2I 
+  1.687 

+  4-  706 

—  0.260 
-  1-540 
-  2.970 

+  0.055 
—  0.067 
+  0.700 

+  1. 

+  0.355 

+  0.152 

-  1.885 

+  0.399 

+  1.224 

+  I.  680 
+  4.703 

-  1.448 
—  2.930 

—  0.087 
+  0.692 

+  O.O22 

+  I. 

+  1-373 

-1.183 

—  O.O7I 

+  2.396 

-  0.945 

+  0.8II 
+  O.OO6 

+  I. 

-0.394 

+  0.338 

+  0.274 

Ex.  i.  Adjust  the  observed  differences  of  longitude*  given  in  the  follow- 
ing table : 

FOILHOMMERUM 
CONTENT 


Fig.36 


Dates. 
1851 

h. 

Cambridge-Bangor,                          o 

Observed  Differences. 
»«.       s.                  ». 
9     23.080  ±  0.043 

1857 

Bangor-Calais, 

6 

00.316  ±  0.015 

1866 

Calais-Heart's  Content, 

55 

37-973  ±  0.066 

1866 

Heart's  Content-Foilhommerum,  2 

5i 

56.356  ±  0.029 

1866 

Foilhommerum-Greenwich, 

4i 

33.336  ±  0.049 

1872 

Brest—  Greenwich, 

17 

57.598  ±  O.O22 

1872 

Brest-Paris, 

27 

18.512  ±  0.027 

1872 

Greenwich-Paris, 

9 

2I.OOO  ±  0.038 

1872 

St.  Pierre-Brest,                                3 

26 

44.810  ±  0.027 

1872 

Cambridge-St.  Pierre, 

59 

48.608  ±  O.O2I 

1869-1870 

Cambridge-Duxbury, 

i 

5O.I9I   ±  O.O22 

1870 

Duxbury-Brest,                                 4 

24 

43.276  ±  0.047 

(1867 
<I872 

Washington-Cambridge, 

23 

41.041  ±  o.oiS 

1872 

Washington-St.  Pierre,                   i 

23 

29-553  ±  0.027 

*  Coast  and  Geodetic  Survey  Report, 

1880, 

app.  No.  6. 

Corrections. 


APPLICATION   TO   TRIANGULATION. 


297 


[Number  of  conditions  =  w  —  s  +  i,  where  n  is  number  of  observed  dif- 
ferences of  longitude,  and  s  is  number  of  longitude  stations. 

The  condition  equations  are 

i. 

—  V»    +V^    —  7'g    =  +  0.086 

—  7'i  —  T'a  —  7>3  —  V  4  —  Z>5  +  Vt     +  Vg     +  Vio  =   +  0.045 

—  Vg  —  V\n  +  V\\  +  Vu  —  —  0.049 

+  Vio  +  7'13  —  Vn  =  —  0.096 

The  weights  are  taken  inversely  as  the  squares  of  the  p.  e. 
Solution  by  method  of  correlates,  as  in  Art.  no.] 

Ex.  z.  The  system  of  triangulaiion  shown  in  the  figure  was  executed  by 
Koppe  in  the  deter- 
mination of  the  axis 
(Airolo  -  Goschenen) 
of  the  St.  Gothard 
tunnel.*  In  the  fol- 
lowing table  the  ad- 
justed values  are  giv- 
en side  by  side  with 
the  measured  values. 
It  is  proposed  as  a  QOSCH 
problem  of  adjust- 


lent. 
At  Goschenen. 

Measured. 

VII                 VI 

Adjusted. 

At  IV. 

Measured.        Adjusted. 

II. 

0° 

oo' 

oo". 

OO 

oo". 

oo 

V. 

O° 

oo' 

oo".oo 

oo".oo 

III. 

44° 

33' 

10". 

88 

10". 

03 

VI. 

15° 

41' 

3"-57 

6".  29 

IV. 

69° 

30' 

12" 

5i 

n". 

62 

VII. 

74° 

1  2 

20".  55 

19".  86 

V. 

124" 

58' 

4" 

23 

5". 

13 

Goschenen 

80° 

32' 

48".  99 

SO".  12 

II. 

135° 

44' 

49"-77 

50".  91 

At  II. 

III. 

199° 

24' 

ii".56 

io".73 

III. 

o" 

oo' 

oo". 

oo 

oo". 

00 

A  *    17 

At  V. 

IV. 

37° 

53' 

54" 

•33 

52". 

97 

IV. 

o° 

oo' 

oo".oo 

oo'.oo 

V. 

60° 

29' 

33" 

•13 

33" 

.82 

VIII. 

78° 

40' 

s'-g1 

6".  72 

VI. 
Goschenen 

77" 
93° 

4' 

n' 

5" 
4i" 

.67 
.69 

8".  17 
40".  5  7 

IX. 
VI. 

140° 
215° 

44' 

32' 

43"-5i 
45"-  4i 

44"-45 

43"-  45 

VII. 

124" 

16' 

33" 

.98 

33".  27 

VII. 

286° 

19' 

25".  30 

27".  21 

At  III. 

Goschenen 

316° 

oo' 

44".  92 

43"-  61 

VIII. 

o° 

oo' 

oo" 

.00 

oo".oo 

II. 

338° 

20' 

33"-53 

3i"-74 

IX. 

53° 

58' 

14" 

.48 

15" 

•49 

At  VII 

VI. 

99° 

47' 

5o" 

.21 

50" 

.86 

II. 

o" 

oo' 

oo".oo 

oo'.oo 

IV. 

102" 

32' 

5i" 

•36 

51" 

.90 

III. 

T9° 

n' 

58".  44 

59"-°3 

Goschenen 

138° 

44' 

28" 

.Si 

29" 

.70 

IV. 

32° 

4' 

49".  32 

4S".6S 

VII. 

I44~ 

28' 

12" 

•47 

Jl" 

.40 

V. 

64° 

n' 

54"-oS 

56".  05 

II. 

i  So0 

59' 

38" 

•94 

39" 

.n 

VI. 

90° 

05' 

39"-47 

37".  oo 

*  Zcitst.hr.  fur  I 

'er»iess.<  vol.  iv. 

298 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


At  VIII. 

At  XI. 

XI. 

o° 

oo' 

oo" 

.00 

oo" 

,00 

XII. 

o° 

oo' 

oo".oo 

oo".oo 

XII. 

18° 

56' 

17" 

•43 

17" 

54 

Airolo 

16° 

55' 

55".o6 

54"-38 

X. 

43° 

50' 

24" 

•  03 

24", 

70 

IX. 

37° 

13' 

59"-  79 

58".43 

IX. 

50° 

18' 

22" 

•  52 

20", 

27 

VIII. 

152° 

26' 

30".  24 

30".  44 

VI. 

106° 

3o' 

15" 

.04 

15" 

37 

V. 

112° 

28" 

.72 

29", 

24 

III. 

130° 

n' 

30" 

.81 

4i". 

54 

At  XII 

At  IX. 

IX. 

o° 

oo' 

oo".oo 

oo".oo 

VI. 

o° 

oo' 

oo" 

.00 

oo", 

,00 

Airolo 

30° 

31' 

2".  30 

3"-  39 

V. 

8° 

28' 

17" 

•  13 

15". 

,06 

X. 

42° 

13' 

20".  53 

2i"-33 

III. 

18° 

33' 

3" 

•  27 

5" 

.00 

VIII. 

90° 

3' 

2".  22 

i  ".74 

VIII. 

63° 

4i' 

28" 

.63 

28". 

55 

XI. 

98° 

40' 

i4"-95 

13".  72 

X. 

76° 

59' 

5o" 

.89 

5i°. 

48 

XL 

79° 

10' 

36" 

•33 

36". 

34 

At  Airolo. 

Airolo 

109° 

45' 

39" 

•23 

39"' 

33 

XII. 

123° 

16' 

23" 

.76 

24". 

23 

XI. 

o° 

oo' 

oo".oo 

oo".oo 

XII. 

94° 

54' 

56".  06 

55".  26 

At  X. 

IX. 

230° 

53' 

7".  5i 

6".  98 

XII. 

o° 

oo' 

oo" 

.00 

oo". 

oo 

X. 

296° 

26' 

49"-43 

51".  n 

Airolo 

9° 

49' 

30" 

.02 

37". 

92 

IX. 

9i° 

30' 

5" 

.16 

5" 

.96 

VIII. 

252° 

43' 

46" 

-75 

47" 

•49 

XI. 

275° 

12' 

8" 

•44 

9" 

•  74 

The  distance  X-XII.  is  44i6»«.8. 

There  are  19  angle  equations  and  15  side  equations  in  the  adjustment. 


SOLUTION  BY  SUCCESSIVE  APPROXIMATION. 


141.  The  rigorous  forms  of  solution  which  have  been 
given  are  suitable  for  a  primary  triangulation  where  the 
greatest  precision  is  required.  In  secondary  or  tertiary 
work  it  would  not  be  advisable  to  spend  so  much  labor  in 
the  reduction,  since  the  systematic  errors  remaining  would 
probably  largely  outweigh  the  accidental  errors  eliminated. 
For  work  of  this  kind  the  method  of  solution  by  successive 
approximation  is  to  be  preferred. 

The  principle  underlying  the  process  of  solution  is  that 
explained  in  Art.  115.  Each  condition  or  set  of  conditions 


APPLICATION   TO   TRIANGULATION.  299 

is  adjusted  for  independently  in  succession,  the  values  of 
the  corrections  found  at  each  adjustment  being  closer  and 
closer  approximations  to  the  final  values.  Should  the 
values  found  after  going  through  all  of  the  conditions  not 
satisfy  the  first,  second,  .  .  .  groups  of  condition  equations 
closely  enough,  the  process  must  be  repeated  till  the  re- 
quired accuracy  is  attained. 

To  make  the  operation  as  simple  as  possible  let  us  take 
but  a  single  condition  at  a  time. 

(i)  Local  equation  at  N.  Base, 


The  solution  is  given  in  Ex.  i,  Art.  121, 

"i\  =    —  o".64,  ?',  =    -  o".64,  7'3  =    —  o".O9 

(2)  Local  equation  at  S.  Base, 

''4  +  ?'.-    7',  +    1.07=0 

The  solution  is  given  in  Ex.  2,  Art.  121, 

v4  =    -  o".i3,  v,  =   -  o".so,  TV,  =:  +  o"-44 

(3)  Angle  equation, 

?'3  +  ?'4  +  *',  +  0.48  =  0 

Using  the  values  of  v3,  v4  already  found  as  first  approxi- 
mations, the  equation  reduces  to 

<;3  +  *'<  +  ?'T+  0.26  =  0 

The  method  of  solution  is  given  in  Ex.  2,  Art.  no, 

*',  =  —  o".i3,  7'4  =    -  o".o8,  r.  =    -  o".os 
39 


THE  ADJUSTMENT  OF  OBSERVATIONS. 

The  successive  approximations  found  so  far,  when 
added,  give 

v,=    -o".64  v6=    -C/.55 

V3  =  —  O".22  V7  =  —  C/.05 

Vt=-Off.2I 

Proceed  similarly  with  the  remaining  two  condition  equa- 
tions. The  resulting  values  will  agree  closely  with  the 
rigorous  values  already  found. 

142.  In  order  to  bring  out  still  more  clearly  the  advan- 
tages of  solving  in  this  way,  let  us  take  a  more  extended 
example.  A  good  one  is  furnished  by  the  triangulation 
(1874-1878)  of  the  east  end  of  Lake  Ontario,  omitting  the 
system  around  the  Sandy  Creek  base. 


.38 


The  measured  values  of  the  angles  are  given  in  the 
following  table.  Each  angle  is  taken  to  be  of  the  same 
weight.  In  the  last  column  are  given  the  locally  corrected 
angles  found  by  the  rigorous  methods  of  solution. 


APPLICATION   TO    TRIANGULATION. 


301 


Station 
occupied. 

Angle  as  measured  between 

Locally 
corr. 
angles. 

Sir  John, 

Carlton  and  Kingston, 
Wolfe  and  Kingston, 

90°     17'     44".  91 
56"     24'     09".  77 

Carlton, 

Wolfe  and  Sir  John, 
Kingston  and  Sir  John, 

120°     48'     06".  54 
62°     03'     27".  56 

Kingston, 

Sir  John  and  Wolfe, 
Carlton  and  Wolfe, 
Wolfe  and  Amherst, 

64°     40'     50".  9  1 
37°     02'     04".  43 
88°     19'     14".  70 

Wolfe, 

Duck  and  Carlton, 
Amherst  and  Carlton, 
Kingston  and  Carlton, 
Sir  John  and  Carlton, 

188°     07'     iH".54 
140°     12'     34".  44 
84°     13'     14".  34 
25°     18'     i6".So 

Amherst, 

Kingston  and  Wolfe, 
Kingston  and  Duck, 
Wolfe  and  Duck, 
Grenadier  and  Duck, 
Duck  and  Vanderlip, 
Vanderlip  and  Kingston, 

35°     41'     23".  02 
in0     45'     28".  46 
76°     04'     06".  32 
54°     38'     oo".34 
71°     15'     25".43 
176°     59'     06".  1  1 

22".  69 
28  ".68 
05  "-99 

25\32 

06".  oo 

Duck, 

Oswego  and  Vanderlip, 
Vanderlip  and  Amherst, 
Amherst  and  Wolfe, 
Wolfe  and  Grenadier, 
Grenadier  and  Stony  Pt., 
Stony  Pt.  and  Oswego, 

104°     08'     58".  93 
70°     26'     3  1  ".99 
56°     01'     12".  47 
18°     45'     43"-36 
49°     53'     12".  77 
60°     44'     19".  46 

59".  10 
32".  16 

12".  64 

43".  53 
12".  94 
19".  63 

Grenadier, 

Stony  Pt.  and  Duck, 
Duck  and  Amherst, 
Duck  and  Stony  Pt., 
Amherst  and  Stony  Pt., 

78°     13'     33".  64 
50°     35'     04".  28 
2SI3     46'     25".  89 
231"     n'     22".  04 

33".84 
04".  19 
26".  16 
21".  97 

Stony  Pt., 

Oswego  and  Duck, 
Duck  and  Grenadier, 
Grenadier  and  Duck, 

88°     22'     oo".S6 
51°     53'     12".  60 
308'     06'     47  ".2  1 

12".  70 
47"-30 

Oswego, 

Sodus  and  Vanderlip, 
Sodus  and  Duck, 
Sodus  and  Stony  Pt., 
Vanderlip  and  Duck, 
Vanderlip  and  Stony  Pt., 
Duck  and  Stony  Pt., 

80°     29'     46".io 
107"     19'     03".  28 
138'     12'     49"-44 
26°     49'     i6".6i 
57°     43'     oi  ".96 
30°     53'     42".SS 

46"-  59 
03".  96 

48'.  28 

17".  37 
oi  ".69 

44"  -32 

Vamlerlip, 

Amherst  and  Duck, 
Amherst  and  Oswego, 
Duck  and  Oswego, 
Duck  and  Sodus, 
Oswego  and  Sod  us. 
Sodus  and  Amherst, 

38       18'     07°.  12 
87"     19'     53"-47 
49°     oi  '     45  ".54 
87°     59'     i2\55 
38"     57'     26".  55 
233'     42'     40".  41 

07  "-30 
53".  16 
45"-  86 
12".  42 
26".  56 
40".  28 

Sod  us, 

Vanderlip  and  Oswego,              60°     32'     57".  55 

3O2  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  local  and  general  equations  are  formed  as  usual  (see 
Arts.  117-140).  The  general  rule  in  the  solution  is  to 
adjust  for  one  condition  at  a  time.  Instead,  however,  of 
following  out  this  rule  strictly,  it  is  often  better  to  adjust  for 
a  group  of  conditions  simultaneously.  Often  a  group  is 
almost  as  easily  managed  as  a  single  condition.  No  rule 
can  be  given  to  cover  all  cases,  and  much  must  be  left  to 
the  judgment  and  ingenuity  of  the  computer. 

143.  The  Local  Adjustment   tit  Each   Station.— 

(a)  Adjust  for  each  sum  angle  separately. 
Rule  and  example  in  Art.  121. 

(b)  Adjust  for  closure  of  the  horizon. 
Rule  and  example  in  Art.  121. 

At  stations  Sir  John,  Carlton,  Kingston,  Wolfe  there  are 
no  local  conditions,  and  at  each  of  the  stations  Amherst, 
Stony  Point,  Sodus  there  is  one  angle  independent  of  the 
others,  and  therefore  not  locally  adjusted. 

The  angles  at  station  Amherst  may  be  rigorously  ad- 
justed, as  in  Art.  121.  The  resulting  values  are  given  in 
the  table.  If  we  break  the  adjustment  into  two  parts,  as  in 

(a)  and  (b),  we  have : 

• 

(a)  Sum  Angle. 

Adjusted. 

22". 73 
06  ".03 

28". 76 

28". 75  check. 


22  ".65 
05  ".96 

25"- 35 
06". 04 


Mea 

sured  values. 

Kingston-Wolfe, 

35° 

41' 

23". 

02  — 

o" 

.29 

Wolfe-Duck, 

76° 

04' 

06", 

32- 

o' 

.29 

iu° 

45' 

29". 

34 

Kingston-Duck, 

111° 

45' 

28". 

46  + 

o" 

.29 

3)o" 

.88 

o" 

.29 

(b)  Closure  of  Horizon. 

Kingston-  Wolfe, 

35° 

4i' 

22". 

73- 

o" 

.08 

Wolfe-Duck, 

76° 

04' 

06". 

03- 

o" 

.07 

Duck-Vanderlip, 

7i° 

15' 

25"- 

43- 

o" 

.0& 

Vanderlip-K  ings  ton, 

176° 

59' 

06", 

ii  — 

o" 

.07 

4)00" 

•  30 

oo" 

•075 

oo".oo  check. 


The  adjusted  values   agree    closely  with   those   from  the   simultaneous 
solution,  as  given  in  the  table. 


APPLICATION   TO   TRIANGULATION.  303 

At  station  Duck  the  angles  close  the  horizon.  Hence 
the  correction  to  each  angle  is  one-sixth  of  the  difference  of 
their  sum  from  360°.  (See  Art.  121.) 

144.  The  General  Adjustment. — The  local  adjust- 
ment being  finished,  we  shall  consider  the  adjusted  angles  to 
be  independent  of  one  another  and  to  be  of  the  same 
weight.  We  are  therefore  at  liberty  to  break  up  the  net 
into  its  simplest  parts.  We  have  in  our  figure,  first  a 
quadrilateral  SCWK,  next  two  single  triangles  KWA, 
AWD,  next  a  central  polygon  DAGSOV,  and,  lastly,  a 
single  triangle  VOS.  These  three  figures  include  most 
cases  that  arise  in  any  triangulation  net. 

(a  i)  Adjustment  of  a  Quadrilateral. — In  the  quadrilateral 
SCKW  all  of  the  eight  angles  i,  2,  ...  8  are  supposed  to 
be  equally  well  measured. 

(i)  The  Angle  Equations. 

The  angle  equations  from  the  triangles  SCIV,  CWK, 
WKS  may  be  written  in  general  terms 

^  +  V,  +  ^3  +  V*   =    A 
^3  +  ^4  +  ^  +  *'*    =   4 

v*  +  *'«  +  *;7  +  v*  =  l* 

As  these  equations  are  entangled,  if  we 
adjusted  for  each  in  succession  a  g»-eat 
many  repetitions  of  the  adjustment  would 
be  necessary  to  obtain  values  that  would 

satisfy  the  equations  simultaneously.     It  is,  therefore,  better 

to  adjust  simultaneously,  and  it  happens  that  a  very  simple 

rule  for  doing  this  can  be  found. 

Call  /(',,  &,,  &,,  /'4  the  correlates  of  the  equations  in  order; 

then  the^  correlate  equations  are 

k,  =  v,  6,  +  £3  =  v, 

kl           =  v,  £,  -f-  /-,  =  vt 


304  THE  ADJUSTMENT   OF   OBSERVATIONS. 

and  the  normal  equations 

4*.  +  2*.  =4 

2k,  -f  4£a  +  2£,  =  4 

2/C-2  +  4^3  =  /3 
Solving  these  equations,  there  result 


Substitute  these  values  in  the  correlate  equations,  and 
»,  =  ».  =  *(+  34  -*4  +   0 

».  =  «'4  =  i(+     4  +  2/,-     /,) 
».  =  ».  =  *(-     4  +  2/a  +     /,) 

»,  =  ».  =  *(+   A-2/a-f-3/,) 
which  ma     be  written 


whence  follows  at  once  the  convenient  rule  for  adjusting 
the  quadrilateral,  so  far  as  the  angle  equations  are  con- 
cerned : 

(«)  Write  the  measured  angles  in  order  of  azimuth  in  two 
sets  of  four  each,  the  first  set  being  the  angles  of  SCW,  and  the 
second  those  of  WKS. 

(,9)  Adjust  the  angles  of  each  set  by  one-  fourth  of  the  dif- 
ference of  this  sum  from  180°  -f-  excess  of  triangle,  arranging 
the  adjusted  angles  in  two  columns,  so  that  the  first  column  ivill 
show  the  angles  of  SCK,  and  the  second  those  of  CWK. 

(y)  Adjust  the  first  column  by  one-fourth  of  the  difference  of 
its  sum  from  1  80°  -f-  excess  of  triangle,  and  apply  the  same  cor- 
rection, with  the  sign  changed,  to  the  second  column. 


APPLICATION   TO   TRIANGULATION. 


305 


The  spherical  excesses  of  the  triangles  SCW,  CWK,  WKS  being  o"i6, 
o".35,  and  o"47  respectively,  the  adjustment  of  (he  quadrilateral  may  be  ar- 
ranged as  follows : 


Measured  angles. 

Adjusted  angles. 

33°  53'  35"-i4 
62°  03'  27".  56 
58°  44'  38".  98 
25°   18'   i6".8o 

35".  56 
27"-  98 
39  "-40 
17".  22 

35"-  4« 
27".  82 
39"-  56 
I?"-  38 

179°  59'  58".4S 
180  +  F.  =  180°  oo'  oo".r6 

oo".i6  check 

4)1".  68 

58°  54'  57"-54 
37     02'  04".43 
27°  38'  46".  48 
56°  24'     9".  77 

58".  1  1 
04".  99 
47".  04 
io".33 

58".  27 

05  "-1  5 
46".  88 

179°    59'    58".  22 

180  +  £  =  180°  oo'  oo".47 

oo".gi 
oo".2S 

oo".47  check 

4)2".  25 

4)0".  63 

o".56 

o".i6 

(2)  The  Side  Equation. 

Using  the  values  of  the  angles  just  found,  we  next  form 
the  side  equation  with  pole  at  O.     It  is 


sin  OSC  sin  OCW  sin  OWK  sin  OKS 
sin  CWO  sin  WKO  s'mKSO 


=  i 


or  writing  it  in  general  terms  when  reduced  to  the  linear 
form  (see  Art.  129) 

art  +  a.pj  +  art  +  art  +  art  +  art  +  art  +  art  =  /< 

where  ?>/,  ?'/,  .  .  .  are  the  corrections  resulting  from   the 
side  equation. 

Solving  as  in  Ex.  2,  Art.  no,  we  have  the  corrections 


-        ^'., 

\ad\ 


p       ^'ti    •    •    • 

[aa] 


306  THE   ADJUSTMENT    OF   OBSERVATIONS. 

These  corrections  may  be  found  still  more  rapidly  as 
follows  :  Since  the  side  equation  may  be  so  transformed 
that  the  coefficients  aiy  a.2,  .  .  .  are  approximately  equal  to 
unity  numerically  (see  Art.  131),  we  may  take  each  of  them 
to  be  unity,  and  then 


that  is,  the  corrections  to  the  angles  are  numerically  equal,  but 
are  alternately  -f-  and  —  . 

This  plan  has  the  additional  advantage  of  not  disturbing 
the  angle  equations. 

Returning  to  our  numerical  example,  we  first  reduce  the  side  equation  to 
the  linear  form. 

OSC    =  33°  53'  35"-40  +  7/1  SCO     =  62°  03'  27".  82  +  v, 

OCW  =  58°  44'  39".  56  +  7/3  CWO  =  25°  18'  17".  38  +  7/4 

OWK=*$°  54'  58".  27  +  7/5  WKO  —  ^  02'  O5".i5  +  z/g 

OA'S    =27°  38'  46".  88  +  2'7  A'SO    =  56°  24'   io".i7  +  7/8 


9.7463587  +  3I-3  »i 
9.9318952  +  12.87/3 
9.9326832  +  12.72/5 
9.6665301  +  40.27/7 

72 
70 

9.9461673  +  1  1.  2  7/a 
9.630869!  +  44.52/4 
9.7797125  +  27.97'e 
9.92O6l8l   +   I4.O7/e 

70 

2 

Dividing  by  20,  which  will  reduce  the  coefficients  to  unity  approximately,  and 
1.567/1'  —  0.567/2'  +  0.647/3'  —  2.222/4'  +  0.642/5' 

—  1.407/d'  +  2.OI7/7'  —  O.7O.7/e'  +  O.IO—  O 

Hence 

[aa]  =  15 
and 

zV  =  —  p".oi,  W  =  o".oo,  7/3'  =  o".oo,  z/4'  =  +  o".oi,  etc. 

By  the  second  rule  the  corrections  would  be  T  —~,  that  is,  To".oi  alternately, 

8 

which  values  differ  but  little  from  the  preceding. 

The  total   corrections  to  the  angles  are  the  sums  of  the  two  sets  of  cor- 
rections from  the  angle  and  side  equations. 


APPLICATION   TO   TRIANGULATION.  307 

(a  2)  Adjustment  of  a  Quadrilateral.  —  By  the  following 
artifice  the  quadrilateral  may  be  rigorously  adjusted  for  the 
side  equation  without  disturbing  the  angle  equation  adjust- 
ment, which  amounts  to  the  same  thing  as  the  simultaneous 
adjustment  of  the  angle  and  side  equations. 

Suppose  that  the  angle  equations  have  been  adjusted  as 
already  explained  in  (a  i).  If  v/,  z'/,  .  .  .  •?'/  denote  the 
corrections  arising  from  the  side  equation,  the  condition 
equations  may  be  written 


",*','  +  <w'  +  "37V  +  <w'  +  <*•?*  -f  <***'« 

B     writin     the  corrections  in  the  form 


the  first  three  condition  equations  become  0  =  0  identi- 
cally, and  we  have  therefore  to  deal  only  with  the  single 
condition  equation 

("i  +  <*9  +  a»  +  "n  —  ^:,  —  ^  —  a.  —  ar)v  -f  (rt-,  —  rt-.j7'' 

+    (",    —    <><'"    +    (^5    —    «B)f'"'   +    (",     —     ^)''""     =     /,' 

with 

(T'  +  T'7+(z'-702+(-7-  +  7'"/  +  (-7'-7'"r+.    .    .    =  a  mill. 

The  correlate  equations  are 


40 


308  THK    ADJUSTMKNT    OF    OBSERVATIONS. 

Substitute  in  the  condition  equation,  and 

k\\(a,  -f  a.,  -f  a:>  -f  a,  —  ay  —  a,  —  rf.  —  a^  +  (>, 


from  which  /•  can  be  found. 

Hence  the  corrections  are  known. 


The  complete  adjustment  of  our  quadrilateral  is  contained  in  the  follow- 
ing table  : 


Meas.  Angles. 

Local 
Angles. 

l."R.  Sines. 

LOR.  Diff. 

1 

o* 

33°   53'  35"-M 

3?",6 

^s"-40      9-74^3587 

+  31-3 

t>2°    03'    27".  56 

27".u8 

9.9461673. 

-f-   I  1.  2 

42-5 

1806 

58°    44'    38"   98 

39'-4° 

39".  56     ;       9.9318952 

+   12.8 

25°  18'  i6".8o 

17".  zz 

i7"-33 

9.6308691 

+  44-5 

57-3 

3283 

58".  48 

oo".i6 

i".68 

580  54'   57"-54 

58".  II 

58"-27 

9.9326832 

+  '2.7 

37°  02'  04".  43 

04  ".99 

05"-'S 

9.7798125 

+  27-9 

40.6 

1648 

27°  38'  40".  48 

47'  -°4 

46".  88 

9.6665301 

+  40.2 

56°   24'   09".  77 

i°"-33 

io".i7 

9.9206181 

+  M-o 

54-2 

2938 

58".  22 

°°',''9o 

—                   — 



oo  '.47 

oo    .28 

^2             7o 

I,!2   5             92.1 

2".  25 

o".63 



2 

4)10.4 

2.6 

5-2 

27 

9702 

42-5       57-3 


2 
9702 


Hence   the  corrections  are  kno-.vn.     These  corrections,  applied  to  the  local 
angles,  give  the  final  angles  required. 

(b)  Adjustment  of  a  Single  Triangle. 

Rule  and  example  in  Kx.  2,  Art.  109,  and  Ex.  2,  Art.  1 10. 

The  single  triangles  in  our  figure  are  Kingston,  Wolfe, 
Amlierst;  Wolfe,  Duck,  Amherst ;  Vanderlip,  Oswego, 
Sodtis. 


APPLICATION   TO   TRIANGULATION. 


309 


For  example,  take  the  first  (.'  =  n".~2) 

Measured.  Adjusted. 

Kingston,        88        19'      14". 70  15". 78 

Wolfe,              55       5<y     20".  io  21  ".18 

Amherst,         35       41'     22". 69  23".  77 

I7<)       5'V     57 '-4'J  o". 73  check. 
180  +  «c  =  I  So"     oo'     oo  .72 


3)3".  23 

r.os 


(c)  Adjustment  of  a  Central  Polygon.  —  In  the  central  poly- 
gon Duck,  Amherst,  Grenadier,  Stonv  Point,  Oswego,  Yan- 
derlip  tlie  condition  equations  in  general  terms  are: 


Fig.40 


Local  equation  (hori/on  equation), 

'u  +  '•„  +  '•»  +  ''„  +  ''„  =  /, 
Angle  equations, 

'      P,  +  <  ',  +  v,  =  1, 
•'       *>       *    =/ 


1  1:1  "l      '  11  "l      '  )r>        '      6 


Side  equation  (j»ole  at  Duck), 


We  may  adjust  tor  these  equations  in  order,  Hist  the  hori- 
zon equation,  then  the  angle  equations  separately,  as  they 
are  not  entangled,  and  next  the  sii.le  equation. 

A  rigorous  adjustment  may,  however,  be  carried  out  at 
once  with  very  little  additional  labor.  Adjust  first  each 
angle  equation  by  itself,  and  let  (:•,},  I;1..),  ...  be  the  values 
that  result.  Let  (l),  (2),  .  .  .  denote  the  farther  correc- 
tions to  the  measured  angles  in  order  arising  from  the  local 
and  side  equations,  so  that 


310  THE   ADJUSTMENT   OF   OBSERVATIONS. 

If  we  substitute    these  values    in    the    above    equations   we 
have  the  new  condition  equations 


(i)  +    (2)4-    (3)=o 
(4)  +    (5)+    (6)=o 

(13)  +  (14)  -|-  (i  5)  =  o 

from  which  to  find  (i),  (2),  (3),  .   .   .  (15). 

Calling  klt  £,,  /,  //,  .   .  .   the   correlates  of  the  condition 
equations  in  order,  we  have  the  correlate  equations 

(1)  =  «,*,  +  7  (4) 

(2)  =  «,*>  +  7  (5) 

(3)=     *  +  7  (6)=     £,  +  77 

Eliminate  now  the  angle  equation  correlates.      By  addition 
(«,  +  ",X'i  +  37 


Hence 


(2)=    -Htf.-w.)^.-i 

(3)  =-!(«,+   «,)*,  +  I 


Substitute    these   values  of  (i),  (2),  ...    in  the    condition 
equations,  and  we  have  the  normal  equations 

2 1  [aa]  —  a,at  —  a<ab  —  .   .   .  JX-,  —  [ii\k9  =  3/' 

-  [a]*,  -f  io/',  =  3/" 

Solving,  we  find  /•,,  &,,  and  thence  (i),  (2),  .   .   .  are  known. 


APPLICATION   TO   TRIANGULATION.  311 

The  normal  equations  may  be  written  down  directly  in 
every  case,  as  the  law  of  their  formation  is  as  evident  as 
that  of  ordinary  normal  equations.  They  involve  only  two 
unknowns,  /',,  /•„,  no  matter  how  many  sides  the  polygon 
has. 

It  is  evident  that  the  elimination  of  the  angle  equation 
correlates  /,  //,...  from  the  correlate  equations  before 
forming  the  normal  equations  amounts  to  the  same  thing  as 
first  forming  the  normal  equations  in  the  usual  way  and 
then  eliminating  /,  //,...  For  the  normal  equations  from 
the  correlate  equations  are 

\aa\k,  +  (a,  +  «,)/  +  (at  +  *.)  //+  .   .    .  =  /' 

+  5*.  +    /  +    //+...=/" 

(«,  +  «#,+    *,  +  3/  =0 

(a4  +  a>)£l+    *,  +3/7  =o 


Substitute  for  /,  //,...  their  values  from  the  third, 
fourth,  .  .  .  equations  in  the  first  two,  and  we  find  the  two 
normal  equations  as  before. 

The  introduction  of  the  method  of  eliminating  the  corre- 
lates arising  from  one  set  of  condition  equations  before 
forming  the  normal  equations  is  due  to  Schleiermacher,  of 
the  Hessian  survey.  For  a  fuller  account  of  it  see  Fischer's 
Geodiisie,  part  iii.  p.  93  ;  Huge  I  in  General  Bt'richt  der  euro- 
pa  isc  hen  Gradinessung,  1867,  pp.  106  seq.  ;  Nell  in  ZcitscJir. 
fiir  Venncss.,  vol.  x.  pp.  I  seq.  ;  vol.  xii.  pp.  313  seq. 

The  process  is  not  of  any  special  advantage  except  in 
such  problems  as  that  under  discussion,  and  then  if.  is 
better  to  use  the  final  formula  for  the  normal  equations 
directly. 

We  shall  now  proceed  with  our  numerical  example. 

At   station    Duck    the    measured    values    of  the    angles 

o 

are  taken,  at  the  other  stations  the  locally  adjusted 
angles. 


312 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


Given  Angles. 

l,og.  Sines. 

Did.  i". 

Squares. 

Products. 

Sums. 

54°  38'  oo".34       oo".62 

9.911  4060 

4-  14-9 

222.  0 

257-8 

-    2.4 

50°   35'  04".  19       04".  47 

9  8879338 

+  '7-3 

299  .  3 

74°  46'  55"-83       56".  12 

180°  oo'  oo"-36 

180°  oo'  oi".zi  =i8o-(-  e 

3)°"-85 

0".28 

78°   13'   33".  84       34".  49 

9:9907654 

+    4.4 

19.4 

72.6 

—    12.  1 

51°  53'   12".  70       13".  35 

9.8958618 

+  16.5 

272.2 

49°  53'   12".  77       13"  42 

i"-95 

o    .65 

88°  22'  oo".86       oo".47 

9.9998235 

+    0.6 

•4 

21  .  I 

-34-6 

3"°  53'  44"-32        43"-93 

9.7105188 

+  35-2 

1239.0 

60°  44     it)".  46       iy".o7 

o4".64 

3  '-47 

°"-39 

26°  49'   17"  37        i8".is 

9.6543842 

+  41.6 

1730.6 

761.3 

+  23.3 

49°  01  '  45".  86       46".  64 

9.8779750 

+  '8.3 

334-9 

104°  08'  58".  93       59"-7° 

2".l6 

4".  49 

a"-33 

o".78 

38°   18'  07".  30       06".  33 

9.7922537 

+  26.7 

712.9 

189.6 

+  19.6 

71°   15'  25'  .32       24".  36 

9  .976^353 

+     7-i 

50-4 

70°  26'  31".  99       3i".o2 

04".  61 

6328                6247 

4881.1 

1302.4 

-    6.2 

i    .71 

6247 

1303.4 

2".  90 

81 

6183.5 

""•97 

3 

2 

243  . 

12367.0 

The  Noimal  Equations. 
12367^,  +  6.2/t,  =  —  243 

6.2^i  +     I0^a  =  2. 01 

.  '.    /'i  =  —  O.O20 

k*  —  +  0.213 


Local  Equation  at  Station  Duck. 

74 '  4»'  5^".  12 
49°  53'  13"- 42 
60  44'  19". 07 
104"  08'  59".  70 
70°  26'  3 1  ".02 

359"     59'     59"-33 
360'     oo'     oo".oo 

oo".f>7 
3 


APPLICATION    TO   TRIANGULATION.  313 

Corrections.  Adjusted  Angles. 


(!)  = 

-o  .39 

54 

3s 

oo  .23 

(2)  = 

+  o".2(> 

5" 

35' 

04"-  73 

(3)  = 

+  o".i3 

74  " 

46' 

5f>"-25 

(4)  = 

-o".24 

7s" 

13' 

34".  25 

(5)  = 

+  o'.iS 

51 

53' 

13".  53 

(6)  = 

+  o".o6 

49° 

53' 

I3'«48 

(7)  = 

—  o".3i 

88° 

22' 

(IT)",  if) 

(8)  = 

-)-  o".4O 

30° 

53' 

44"-33 

(9)  = 

—  o".O9 

60° 

44' 

iS".(>S 

(10)  = 

-o".;5 

26° 

49' 

17".  40 

(")  = 

+  o".45 

49" 

01' 

47"-  09 

(12)  = 

4-  o".3O 

104 

08' 

60".  oo 

(13)  = 

-  o".47 

38C 

1  8' 

05  ".86 

(M)  = 

+  O".20 

71 

15' 

24".  56 

(•5)  = 

4-  0".27 

70° 

26' 

31  ".29 

APPROXIMATE  METHOD  OF  FINDING  THE  PRECISION. 

145.  An  adjustment  may  be  carried  out  rigorously  so 
far  as  finding  the  values  of  the  unknowns  is  concerned,  but 
only  an  approximate  value  of  the  rn.  s.  e.  of  the  angles  or 
sides  may  be  thought  necessary. 

In  good  work  the  following  method  will  give  results 
nearly  the  same  as  those  found  by  the  rigorous  process. 

The  average  value  //  of  the  m.  s.  e.  of  an  angle  in  a 
triangulation  net  after  adjustment  is  easily  seen  from  Art. 
102  to  be 


n  — 


where 

;/  m  number  of  angles  observed. 

nc  =  number  of  local  and  general  conditions. 

ft  —  m.  s.  e.  of  a  measured  angle  of  weight  unity. 

The  value  of//  is,  by  the  usual  formula, 


THE   ADJUSTMENT   OF   OBSERVATIONS. 

To  find  the  m.  s.  e.  of  a  side  of  a  triangle  a  single  chain 
of  the  best-shaped  triangles  between  the  base  rind  the  side 
is  selected,  all  tie  lines  being  rejected.  Then,  assuming  the 
base  to  be  exact  and  the  m.  s.  e.  of  eacli  adjusted  angle  to 
be  //,  we  have  from  Ex.  9,  p.  234, 


where  OA,  OB  are  the  log.  differences  corresponding  to  i"  for 
the  angles  A,  B  in  a  table  of  log.  sines. 

A  form  still  more  approximate  was  used  on  the  U.  S. 
Lake  Survey  in  the  determination  of  the  precision  of  a  side 
of  the  primary  triangulation.  The  angles  ot  each  triangle 
were  taken  to  be  independent  of  one  another.  In  this  case 
evidently  (Ex.  5,  p.  109) 


The  earlier  work  of  the  Coast  Survey  was  computed  from 
this  same  formula. 


Ex.  To  find  the  m.  s.  e.  of  the  side  OL  as  derived   from    the  base  NS  in 
the  figure  ONSL  (Fig.  19). 

Number  of  angles  measured  =  9. 

Number  of  conditions,  local  and  general,  =  5. 

From  the  adjustment  (Art.  140)  [/?>?']  =r  7.54. 


5 


APPLICATION   TO   TRIANGULATION. 
The  chain  of  triangles  is  OArS,  OLS. 


315 


Station. 

Angles. 

5                                Sqs. 

Prods. 

N.  Base, 

122      II'    15" 

-    13-2                        174.2 

S.  Base, 

23^  oS'  05" 

-  401.3 

Oneota, 

34    40'  40" 

+  30-4                        924-2 

S.  Base, 

70°  39'  25" 

+    7-4                   54-« 

Oneota, 

78"  27'  05" 

260.  5 

Lester,                    30°  53'  30"             -+-  35.2              1239.0 

2251.4 

=  31.6  in  units  of  the  seventli  decimal  place. 
Also  OL  —  16556  metres. 


Hence  /inr  is  known. 


The  RIctJiod  of  Directions. 

146.  This  method  is  clue  to  Bessel.  Various  modifica- 
tions of  Bessel's  plan  of  making  the  observations  are  used 
on  different  surveys.  The  following  is  that  used  on  the 
U.  S.  Coast  Survey.  . 

"  In  any  set,  after  the  objects  have  been  observed  in  the 
order  of  graduation  they  are  re-observed  with  instrument 
reversed  in  the  opposite  order:  the  mean  of  the  two  obser- 
vations upon  each  object  is  then  taken.  The  number  of 
such  sets  and  the  number  of  positions  made  depends  on  the 
accuracy  required  and  upon  the  perfection  of  the  instru- 
ment." A  single  series  of  means  is  called  an  "arc." 

"The  direction  instrument  requires  that  it  should  be 
turned  on  its  stand  or  changed  in  position,  in  order  that  the 
direction  of  any  one  line,  and  consequently  of  all,  should 
fall  upon  different  parts  of  the  circle  as  the  only  security 
against  errors  of  graduation.  The  number  of  positions 
41 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


varies  from  five  to  twenty-one  of  nearly  equal  arcs;  and  in 
each  position  the  circuit  of  the  horizon  is  made,  giving  the 
direction  of  each  line  by  two  observations,  one  in  the  direct 
and  the  other  in  the  reversed  position  of  the  telescope. 
These  circuits  or  series  are  repeated  in  each  position  until 
two  to  five  values  of  each  direction  are  obtained.  Each 
angle  is  therefore  determined  by  from  35  to  63  measure- 
ments in  the  direct  and  a  like  number  in  the  reversed  posi- 
tion of  the  telescope." 

147.  The  Local  Adjustment. — Let  O  be  the  station 
occupied,  and  i,  2,  3,  .  .  .  the  stations  sighted  at  in  order 
of  azimuth.  Let  some  one  direction, 
as  Oi,  be  selected  as  the  zero  direc- 
tion, and  let  A,  B,  .  .  .  denote  the 
most  probable  values  of  the  angles 
which  the  directions  of  the  different 
signals  make  with  this  direction. 

In  the  first  arc  let  Xl  denote  the 
most  probable  value  of  the  angle  be- 
tween the  zero  of  the  limb  of  the 
instrument  and  the  direction  of  the 

signal  taken  as  the  zero  direction  ;  then  if  J//,  J//',  .  .  . 
denote  the  readings  of  the  limb  for  the  different  signals, 
and  z//,  v" ,  .  .  .  the  most  probable  corrections  to  these 
readings,  we  have  the  observation  equations 


Fig.4l 


The  zero  of  the  limb  being  changed  in  the  next  arc,  we  have 
in  like  manner 

X  -  M  =  v' 


and  so  on  for  the  remaining  arcs. 


APPLICATION   TO   TRIANGULATION.  317 

If  now  />,',  /,",  .  .  .  ;  //,  /»,",  .../...  denote  the  weights 
of  the  measured  directions  of  the  several  series  of  arcs,  the 
normal  equations  follow  at  once.  They  are 

[>,]  x,  +P:  A  +pin  B+  .  .  .  =  [/vi/j 


p 


from  which  the  unknowns  may  be  found. 

In  order  to  shorten  the  numerical  work  a  course  similar 
to  that  of  Art.  41  may  be  followed.     Let 


where  J//,  Ml,  .  .  .  A',  B ',  .  .  .  are  approximate  values  of 
Xlt  X,  ...  A,  £,...,  and  .f,,  xv  .  .  .  (A),  (B),  .  .  .  denote 
their  most  probable  corrections. 

Also,  for  convenience  in  writing,  put 

ml'   =  Ml'   -  Ml  -  A'  ml'  =  J/a"  -  Ml  -  A' 

ml"  =  Ml"  -  Ml  -  B'  ml"  =  Ml"  -  Ml  -  B' 

The  normal  equations  now  become 


-h 


3l8  THE   ADJUSTMENT    OF   OBSERVATIONS. 

The  quantities  ,flf  x.t,  .  .  .  being  merely  auxiliary  quanti- 
ties, we  eliminate  them  by  substituting  their  values  as  found 
from  the  first  group  of  normal  equations  in  the  second 
group.  We  have  then 


« 


which  may  be  solved  as  usual. 

These  equations  may  be  written 


where    [««],    [rf^]>    •    •    •    are    to    be    looked    on    as    mere 
symbols. 

In  the  cases  that  occur  in  ordinary  work  the  computa- 
tion may  be  still  farther  shortened.  If  we  arrange  the 
observations  in  groups  containing  readings  on  the  same 
series  of  signals,  then,  these  readings  being  of  equal  value, 
we  have  for  the  first  group 

A/=A//  =  ./  suppose 

A'=A"  =,/         " 

. 

and  therefore 


;//  being    the    number   of  signals  sighted  at  in  this  group. 
Similarly  for  the  other  groups. 


APPLICATION   TO   TRIANGULATION. 


319 


Hence  if  ///,  //rt",  .  .  .  denote  the  number  of  arcs  in  the 
several  groups,  the  coefficients  of  the  normal  equations 
become 

//  '  »/'     „'/ 

[<W]  =  L/J-    ->  •/          -*T,*P 

>'s  »s 


where  ,/>',  ,/>"„  .  .  .  denote  any  of  the  equal  weights  in 
the  several  columns  of  the  first  group,  and  „/>',  2/>",  .  •  •/ 
3p',3p",  .../..  .  denote  corresponding  quantities  in  the 
second,  third,  .  .  .  groups.  These  weights  for  the  signals 
sighted  at  may  be  taken  to  be  each  equal  to  unity,  and,  for 
the  signals  not  sighted  at,  zero. 

After  having  found  the  quantities  ///,  w",  ...  by  taking 
the  differences  between  the  approximate  values  of  the 
angles  and  the  several  measured  values,  it  is  convenient  to 
arrange  the  formation  of  the  normal  equations  according  to 
the  following  scheme: 


No.  of 
Group. 


/"'«" 


[P\    [/'"'"I 


The   coefficients  of  the  normal   equations  may  now   be 
written  down  at  sight. 


320  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Special  Case.  —  If  every  arc  is  full,  and    every  direction 
is  equally  well  measured  in  every  arc,  then 


and  the  normal  equations  become 


By  addition,  the  number  of  arcs  being  //s  —  i, 

w 


•  ••)==? 


Hence 

«„(£)  =  [;«'"] 

as  is  evident  a  priori. 

148.    Checks  of  the  Normal  Equations. 
i.  The  sum 


[  cc  _]  +  .   .   . 


is  equal   to  half  the  number  of  observations,  less  half  the 
number  of  arcs. 


APPLICATION   TO   TRIANGULATION.  321 


For  substituting    for    [#<?],  [^],  •   •   •   their  values   we 
find  the  above  sum  equal  to 


—  pi   +  fi"  +  Pi'"  +    .   .    .  _  /»i"2+//j!+/>i"/i"'  4-  .  .  . 

2  [/i] 

—  pi"   -$- pi" -t- ft"  +  .    .    .        /'-•'  2  4  A'"!  4 />2  'A'"  4  ... 


[/»] 

4-    -  ... 

=  1 1  [/]  +  [/"]  +  [/"]  +  -       •  I  ~  ^Jjj  -  ^|  - 

Now,//  — //  ==  .  .  .  =  i,  and  therefore 

[/»']  -f-  [/>"]  -j-  \ p'"]  -}-...        =  the  number  of  observations 

r/,2]       r/,2]       ry,ai 

-^  -f  .  .  .   =  the  number  of  arcs. 


[AJ       [A]        [A] 
2.  The  sum 

[f/]  +  [*/]  +  L^]  +  -  •  •  -[ 

where  [«'/]  is  formed  in  the  same  way  as  [^ 
For 


since  [/"w/"J  +  [/'"'«'"]  +  •  •  • 

-  [A'«.l  -  [AWJ  -...==  [ 

^=  o 
which  proves  the  proposition. 


322  THE   ADJUSTMENT   OF   OBSERVATIONS. 

149.    The  Precision  of  the  Adjusted  Va/ucs.  —  The  first  step 
is  to  find  //,  the  m.  s.  e.  of  a  single  direction. 
We  have  generally 


. 

No.  ol  obs.  quan.  —  No.  of  indep.  unknowns. 

Now,  from  the  observation  equations,  Art.  147,  it  is  evident 
that  the  number  of  independent  unknowns  is  equal  to  the 
number  of  arcs  //„,  together  with  the  number  of  signals  ;/, 
sighted  at,  less  one.  Hence 

\j>w\ 

ft     -  :     -  L/ 

n  —  (tra  -\-us  —  I  ) 

n  being  the  number  of  directions. 

To  compute  the  value  of  |  pvi<\.  Eliminate  .rn  x^,  .  .  . 
from  the  normal  equations,  Art,  147,  and  we  have  the  re- 
duced normal  equations 


+  A  V)  +//"(#)  +  •  •  •     =[A< 


Hence,  as  in  Art.  100, 


A         AJ 

=  [p,nm\  -  Iil-I 


The  quantity  \pnnn\  should  always,  if  possible,  be  found 
from  the  original  observations.  It  will  in  general  be  quite 
different  if  found  from  the  means  forming  the  different 
groups  of  arcs  taken  as  single  observations.  (See  Ex., 
Art.  62.) 


APPLICATION   TO  TRIANGULATION. 


323 


Takinof  the  weight  of  each  observed  direction  to  be  o 

o  o 

the  same  value  unity,  the  expressions  for[z/z']  may  be  written 


\vv]  =  \jnni\  — 


['«,]' 


1^1 

-[aa\ 


150.  The    General    Adjustment. — The    general   ad- 
justment is  carried  out  as  in  the  case  of  independent  angles. 
The  angle  and  side  equations  are  formed  as  in  Arts.  1 17-139, 
and  the  solution  effected  according  to  the  programme  of 
Art.  113. 

151.  Ex.  At  station  Clark  Mt.,  in  the  triangulation  of  the  Blue  Ridge,  Va,, 
readings  were  made  with  a  non-repeating  theodolite  in  the  method  of  arcs. 
The  following,  taken   from  these  readings,  will   be   sufficient  to  illustrate  the 
method  of  reduction. 

The  original  observations  are  arranged  in  sets  containing  readings  on 
the  same  groups  of  signals,  and  the  quantities  given  in  the  table  beiow  are 
the  remainders  found  by  subtracting  the  reading  of  the  first  direction  in  earh 
series  from  the  readings  of  the  other  directions;  that  is,  MI"  — Mi, 
Mi" -Mi,  .  .  . 

Ai  Claik  Mt. 


Spear.                            Humpback 

Fork. 

oo"  oo'  00".  oo 

.OO 
.CO 
.OO 

24°    09'    35^-70 

33"-55 

78'   26'  oS".55 
09  ".60 

09".  33 
io".45 

oo'  oo'  00".  oo 

.OO 

78"'    26'    IO".20 
1  1  ".03 

oo    oo'  oo'.oo 

.OO 
.OO 
.OO 
.00 

24°  og'  36".  10 
37"-4Q 

3S".I5 
39".oo 

oo    oo'  oo'.oo 
.00 
.00 
.00 

54     1  6'  31".  85 
3i"  94 

36'iig 

42 


324 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


The  Local  Adjustment. 
Assume  the  most  probable  values  of  the  angles 

Spear,  00°  oo'  oo".oo 

Humpback,     24°  09'  36". 90  +  (A) 
Fork,  78°  26'     9". 90  +  (£) 

Take  the  differences  between  the  approximate  values  of  the  angles  and 
the  several  measured  values.     We  then  have 


/';«' 

p"m" 

p'"  m'" 

Sums. 

o.oo 

na'  =  4      .00 
.00 
«/  =  3      .00 

o.oo 

—  1.  2O 
i/'  =  1        +   2.50 
—  0.67 

-  3-35 

-  1-35 

i/"r=I        —0.30 
-0.57 
+  0.55 

-4-39 

—  2.72 

-1.67 

1la"=2     O.OO 

ns"—2      .00 

if'1  -  0 

+  0.30 
tp'"  =  I       +  I.I3 

+  1-43 

o.oo 

+  i-43 

o.oo 

—  0.80 

.00 

.00 

.00 
.00 

o.oo 

+  0.50 
—  0.37 

+  1.25 

+  2.IO 

+  2.68 

+  2.68 

o.oo 
.00 

-  1.15 
—  i.  06 

.00 
.00 

-0.97 
+  3-13 

—  0.05 

0.00 

—  0.05 

[/'»/']  =  o.oo 

\_p"m"]  =  -  0.04 

[/'"w'"]=:  —  0.29 

[/»/]  =  -0.33 

[/]  =  II 

[/']  =  J3 

[/"]  =  10 

APPLICATION   TO   TRIANGULATION. 
Next  form  the  table 


325 


No.  of 
group. 

"a 
"s 

"a 

p'  in' 

/";«" 

/'";«'" 

Sum. 

Sum 
ns 

I 

1 

4 

0 

—  2.72 

-1.67 

~4-39 

~  I-463 

2 

a 

2 

o 

+  1-43 

+  1.43 

+  0.715 

3 

f 

5 

0 

+  2.68 

+  2.68 

+  1-340 

4 

^ 

4 

o 

.00 

—  0.05 

—  0.05 

—  0.025 

Sums, 

15 

—  0.04 

—  0.29 

-  0.33 

Check 

The  coefficients  of  the  normal  equations 

[jzfl  ]  =  13  -!-f-!  =  +  7i 
[_**_]  =  -  I  -  1  =  -  34 
[_W_]  =  IO-f-f-f=  +  5f 

r  ,r/  ]  =  —0.04  +  1.463  —  1.34  +  0.025  =  +0.108 
r  bl  i  =  —0.29  +  1.463  —  0.715  +0.025  =  +  0.483 
[^/]  =-1.463  +  0.715  +  1-34  =+0.592 

Check  (i)  [art]  +  [jitfj  +  [_WJ  =  gj 

=  i(34-i5) 


as  it  should. 

(2)  [ 

as  it  should. 

The  normal  equations  are 


=  0.5QI 
=  [«,/] 


=+  0.483  = 

The  general  solution  of  these  equations  gives 
C<4)  =  +  0.1921  [a/]  +  o. 
(B)=  +  O.I  1  30  [a/]  +o.242g[W] 

Substituting  for  [«/],  [^/]  their  values,  there  result 

(A)—  +  o".075 
(B)=  +  o".i30 


326  THE   ADJUSTMENT   OF   OBSERVATIONS. 

and  hence  the  local  directions 

Spear  o°     oo'     oo".ooo 

Humpback,     24°     09'     36".  975 
Fork,  78°     26'     io".O3O 

To  find  the  m.  s.  e.  of  a  single  observation. 

The  value  of  [?'?'],  computed  according  to  Art.  150,  is  found  to  be  25.7. 
Therefore 


the  divisor  in  this  case  being  15. 

This  completes  the  local   adjustment  at  this  station.     Proceed  similarly 
at  the  remaining  three  stations. 

The  General  Adjustment. 

At  Clark. 
Most  probable  directions. 

Spear,  o°     oo'     oo".ooo 

Humpback,    24°     09'     36".975+(i) 
Fork,  78°     26'     io".o3o  4-  (2) 

Weight  Equations. 

(l)=  +  O.I92I  |  I  |  +  O.II30  |  2  | 
(2)=  +O.II30  |  1  |  +  0.2429  |  2  | 

[w]=25.7     Divisor,  17 


At  Spear. 

Humpback,      o"     oo'  oo".ooo 

Fork,                32"     08'  n".793+(3) 

Clark,               54°     06'  29".  197  +  (4) 

Weight  Equations. 

(3)=  +  0.2061  jjj   +0.0485  fjj 
(4)=  +  0.0485  |Tj   +0.1879  IT? 

\vv\  =  58.7  Divisor,  17 


APPLICATION   TO*  TRIANGULATION.  327 

At  Humpback. 

Clark,         o"     oo'     oo".ooo 
Spear,      101°     44'       3". 123 +  (5) 
Fork,       332°     58'     n".i57+(&) 

Weight  Equations. 

(5)=  +0.1333  I  5  I  +0.0667  I  f>  I 

(6)=  +0.0667  |  5  |  +0.1833  |  6  | 

[rT']  —  106.0  Divisor,  23 

At  Fork. 

Clark,  o°     oo'     oo".ooo 

Spear,  79°     35'     42".  479  +  (7) 

Humpback,  98°     41'     43". 926 +  (8) 

Weight  Equations. 
(7)=  +0.2970  |  7  I  +- o.i 394  I' 8  | 
(8)=  +0.1394  Qj  +0.1879  |JJ 
[<vz/]=47.5  Divisor,   17 

The  angle  and  side  equations  are  formed  as  already  explained.  The 
angle  equations  from  the  triangles  SFC  (E  =  io".773),  HFC  (f.  —  7". 386), 
SI/C(f.  —  9". 789),  and  the  side  equation  from  the  quadrilateral  CSHF  (pole 
at  C),  will  be  (ound  to  be 

(2) -(3) +  (4) +  (7) -0.860 

-(i) +  (2) -(6) +  (8)  =  1.562 

(i)  +  (4)  +  (5)  =0.494 

2.609(3)  -  1.847(4)  +  0.2187(5)  -  2.0635(6)  +  0.0193(7)  +  0.1611(8)  =  0.0424 

From  this  point  the  solution  is  carried  through  exactly  as  in  Art.  140. 
The  finally  adjusted  directions  will  be  found  to  be 

At  Clark,   Spear,  o"  oo'  oo".ooo     At  Humpback,  Clark,       o'  oo'  oo".ooo 

Humpb-.ck,  24*  09'  36". 844  Spear,  ioT  44  03". 279 

Foik,  78°  26'  io".47S  Fork,    332'  58'  io".7S4 

At  Spear,   Humpback,    o°  oo'  oo".ooo     At  Fork,      Clark,  o°  oo'  oo'.ooo 

Fork,  32°  08'  n".799  Spear,  79°  35'  42". 428 

Clark,  54"  06' 29". 666  Humpback,  98°  41'  44". 536 

The  m.  s.  e.  of  an  observation  of  weight  unity  is  i'-77. 

The  m.  s.  e.  of  the  adjusted  value  of  the  angle  CSF  =  o".46. 


328  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  form  of  reduction  used  on  the  U.  S.  Coast  Survey 
for  the  adjustment  of  the  primary  triangulation  is  essentially 
the  same  as  that  just  explained,  so  far  as  finding  the  val- 
ues of  the  local  corrections  is  concerned.  The  method  of 
weighting  employed  in  the  general  adjustment  is  not  given, 
as  it  is  not  so  satisfactory  as  that  in  the  text.  It  will  be 
found  in  Report,  1864,  app.  14. 

152.  Approximate  Method  of  Reduction.  —  A  very 
convenient  method  of  approximation  may  be  derived  from 
the  normal  equations,  Art.  147.  It  depends  on  the  theorem 
of  Art.  115,  and  hence,  if  the  process  is  repeated  often 
enough,  leads  to  the  same  result  as  the  rigorous  solution. 

In  the  second  group  of  normal  equations,  Art.  147,  assume 

X,  =  M,',   X,  =  M,',  .  .   . 
then  we  have  as  approximate  values  of  A,  B,  .  .  . 


_  \p"(M"  -  _ 


[/'J  [/"] 

Let  xl  denote  the  correction  to  X^  ,t'2  to  X^  .  .  .  Then  the 
values  A',  />',  ...  of  A,  />,  .  .  .  substituted  in  the  first 
group  of  normal  equations,  give  as  approximate  values  'Of 


_ 


[A] 


which  values  substituted  in  the  second  group  give  as  second 
approximations  to  the  values  of  A,  B,  .   .   . 

A  »  _  A'W:  •  -  M:  -  £/)  +P:(M:-  -  M:  -  Q  +  .  .  . 

T7T 

W'-M';-  Q  +  .  .  . 


and  so  on. 


APPLICATION   TO   TRIANGULATION.  329 

This  approximate  form  of  reduction  was  used  on  the 
Ordnance  Trigonometrical  Survey  of  Great  Britain  in  the 
reduction  of  the  principal  triangulation.  The  approxima- 
tions were  carried  out  only  as  far  as  A",  B\  .  .  .  This  is 
in  general  sufficient  in  good  work. 

Instead  of  finding  A',  Z>',  .  .  .  as  above,  it  is  often  more 
convenient  to  follow  the  Ordnance  Survey  plan,  and,  in- 
stead of  deducting  J//,  AJt",  .  .  .  from  the  readings  of  the 
signals  in  the  different  arcs  in  order,  to  add  to  the  readings 
of  the  signals  in  the  different  arcs  quantities  which  will 
make  the  column  M'  constant  throughout.  We  should 
then  have 


The  corrections  x  and  the  other  approximations  are  made 
as  before. 

Should  the  observation  of  the  zero  direction  be  want- 
ing in  any  arc  —  as,  say,  the  third  —  the  quantity  to  be  added 
to  each  reading  in  this  arc  is  given  approximately  by  the 
value  of  X3  in  the  first  set  of  normal  equations.  Thus,  given 
the  readings  prepared  as  explained  above, 

M',  MS',  M,'" 
M',  M^',  Mtf" 

o,     M,'"  -  M," 
to  find  M:  -  M3f. 

If  A',  B'  have  been  found  from  the  first  two  arcs,  then 
from  the  normal  equations 


and 


M  "  -  M  '- 


=  Ar  approx. 


330  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Hence  the  complete  form  would  be 

M',  M,",  M,'" 

M',  M:,  M,'" 

A'  +  M',  M,'"  -  M3"+Af  +  M' 


or 


Ex. 


o.MS-  Mf,  M/"  -  M' 
o,  M,"  -M',  Mn'"  -M' 

A',  M,'"  -Mt"-\-A' 

At  Clark  Mt. 


p 

M"  -  M' 

M'"  -  M' 

4 

2 

00°  oo'  oo".oo 
.00 

24°    09'    36".  22 

78°  26'  09".  48 
io".62 

5 
4 

.00 

37"-44 
00°  oo'  oo" 

54°   1  6'  32.  "99 

A'  =  24°  09'  36".  90 

4 

2 

00°  oo'  oo".oo 
.00 

24°    Og'    36".  22 

78°  26'  09".  48 

!0".62 

5 
4 

.00 

37"-  44 
36".  90 

9".  89 

Means 

36  ".90 

9".  88 

Corrections  to  the  arcs: 

M"  -  M1  —  A' 

M'"  —  M'  —B' 

X 

o".oo 

-o".6S 

—  o".4O 

—  o".36 

.00 

+  o".74 

+  o".37 

.00 

+  o".54 

+  0".27 

o".oo 

+  o".oi 

+  o".oo 

M'—M'—x 

M'"  —  M'  —  x 

4 

2 

o"-36 
59"-63 

24"  09'  36".  58 

78°  26'  09".  84 

I0".25 

5 
4 

59".  73 

37"-  17 
36".  90 

9".  89 

Means,               59"-94 
Final  values,    oo".oo 

3f>"  -90 
36".  96 

9"-94 
9".  98 

APPLICATION  TO  TRIANGULATION.  331 

Modified  Rigorous  Solution. 

153.  The  forms  that  have  been  given  in  the  preceding 
articles  for  the  rigorous  adjustment  of  a  triangulation, 
though  analytically  very  elegant,  are  somewhat  compli- 
cated. A  method  which  shall  give  a  marked  diminution 
of  work  in  the  reduction  without  increasing  the  field  work 
materially  is  a  desideratum. 

In  the  reduction  of  a  long  net  of  triangulation  we  have 
seen  that  labor  is  saved  by  breaking  the  work  into  two 
parts,  first  adjusting  at  each  of  the  stations  for  the  local 
conditions,  and  then  using  this  work  in  the  further  adjust- 
ment arising  from  the  angle  and  side  equations.  Now,  if 
the  measurements  were  made  on  a  uniform  plan  the  local 
adjustment  would  be  simplified,  as  we  should  have  a  similar 
problem  to  solve  at  each  station.  This  would  lead  to  a 
great  saving  of  labor,  since,  if  the  measurements  are  made 
at  hap-hazard,  the  local  adjustment  may  be  quite  compli- 
cated. 

If  we  decide,  then,  that  the  observer  must  work  in  ac- 
cordance with  some  regular  form,  our  next  inquiry  is,  What 
shall  that  form  be  ?  First,  shall  the  angles  be  measured  in- 
dependently or  in  arcs  ?  The  point  to  be  aimed  at  in  this 
as  in  all  work  of  precision  is  to  get  rid  of  systematic  error. 
The  accidental  errors  are  trifling  in  comparison.  When 
we  consider  twist  of  triangulation  station  from  the  action 
of  the  sun's  rays;  the  influence  on  distinctness  of  vision  for 
the  same  focus  for  different  lengths  of  lines  sighted  over; 
the  interruptions  that  may  occur  in  the  course  of  reading  a 
long  arc;  the  more  uniform  light  that  may  always  be  had 
when  the  number  of  signals  in  use  at  one  time  is  small,  etc., 
we  cannot  but  conclude  that  greater  precision  is  to  be 
attained  by  measuring  the  angles  independently.  The 
errors  are  more  likely  to  mutually  balance.  Even  Andras, 
the  author  of  the  most  important  contributions  to  the 
method  of  directions  since  Bessel,  and  who  used  this 
method  in  the  triangulation  of  Denmark,  acknowledges 
43 


332  THE   ADJUSTMENT   OF   OBSERVATIONS. 

that  "  in  place  of  observations  of  directions  in  arcs  it  is  pre- 
ferable to  return  to  the  old  method  of  Gauss  in  measuring 
angles."  * 

As  regards  the  cost,  it  must  be  acknowledged  that  for 
an  equal  number  of  results,  leaving  quality  out  of  account, 
the  method  of  arcs  has  the  advantage.  Nowadays,  how- 
ever, when  facilities  exist  for  measuring  angles  by  night  as 
well  as  by  day,  there  is  less  delay  in  waiting  for  suitable 
conditions  than  when  day  work  alone  had  to  be  depended 
on.f  Taking  this  into  account,  the  difference  in  cost  would 
not  be  great  in  any  case,  more  especially  as  a  triangulation 
party  is  never  a  very  large  one. 

Having  decided  that  angles  should  be  measured  inde- 
pendently, it  is  in  accordance  with  general  experience  that 
instead  of  spending  all  of  the  time  of  observation  in  meas- 
uring the  single  angles  themselves  better  results  would  be 
obtained  by  spending  part  of  it  in  measuring  combinations 
of  the  angles.  A  simple  form  that  at  once  suggests  itself 
would  be  to  close  the  horizon  at  each  station  ;  that  is,  to 
measure  all  of  the  angles  AOB,  BOC,  .  .  .  LOA  in  order 
round  the  horizon  (see  Fig.  20).  The  local  corrections 

would  each  be  -    of  the   discrepancy    of  the   sum    of  the 

angles  from  360°  (Art.  121),  and  the  reduction  is  thus 
simple  and  uniform.  However,  as  measuring  the  closing 
angle  LOA  is  the  same  as  measuring  the  sum  angle  AOL,  it 
would  seem  that  if  we  measure  one  sum  angle  we  ought  to 
measure  all  possible  sum  angles.:}:  Though  the  form  of 
adjustment  for  this  combination  of  measures  is  a  special 
case  of  that  already  given,  I  shall,  at  the  risk  of  a  little 
repetition,  sketch  it  in  full. 

*  VerhaniUungen  der  europtiischen  Gradmcssung,  1878,  p.  47. 

t  Sec  C.  S.  Report  1880,  App.  No.  8.  Experiments  made  at  Sugar  Loaf  Mountain,  Georgia, 
have  shown  that  an  apparatus  cheap  and  easily  operated  can  be  used  ;  that  night  observations  are 
a  little  more  accurate  than  those  by  day,  and  <that  the  average  time  of  observing  in  clear  weather 
can  be  more  than  doubled  by  observing  at  night. 

t  This  form  of  combination  was  introduced  by  Gauss  in  the  triangulation  of  Hanover. 
A  similar  form  is  employed  on  the  New  York  State  Survey.  See  also  C.  S.  Report  1876, 
App.  20. 


APPLICATION   TO   TRIANGULATION. 


333 


154-  The  Local  Adjustment  (Angles). — Let  O  be  the 

station  occupied,  and  1,2,3,4  the  sta- 
tions sighted  at  in  order  of  azimuth  ; 
then  the  angles  to  be  measured  would 
be 

1 02 

103  203 

104  204  304 

Take   the   first   three   as   independent 

unknowns,  and  let  A,  B,  C  denote  their 

most   probable   values.      Also   let   /ia,  /13,  .  .   .  denote   the 

several  measured  values. 

The  observation  equations  are 


Fig. 43 


-A  +  B 
-A 


and  the  normal  equations 


(0 


Solve  these  equations,  and 


(3) 


It  is  useful  to  notice  as  a  check  that 

A+JS+C=llt  +  itt  +  /lt  (4) 

For  practical   computation   arrange  in  tabular  form  as 
follows.     Sum  first  the  horizontal  rows,  next  form  the  sum 


334 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


differences,  and  then    place   these  quantities  in  the  proper 
columns.     Add  each  column  and  divide  the  sums  by  4 :      . 


A, 

/,. 

/„ 

Sum, 

4s 

4,4 

Sum, 

Sum,  —  Sum2 

A  4 

Sum3 

Sum,  —  Sum3 

Sum,  —  Sum, 

Sum,  —  Sum3 

Sum, 

A 

£ 

C 

Check 

Check  Solution.  —  The  form  of  the  expressions  found  as 
the  values  of  A,  B,  C  suggests  a  method  of  finding  these 
values  in  accordance  with  the  fundamental  principle  of  the 
mean.  For,  writing  the  value  of  A  in  the  form 


A  = 


2+1  +  1 


(5) 


we  see  that  it  is  the  weighted  mean  of  the  values 


weight  i 


- 


that  is,  the  value  of  A  is  found  by  taking  the  weighted 
mean  of  the  measured  value  of  A  and  of  all  the  values  of  A 
that  can  be  found  by  combining  two  other  measures. 

Similarly  for  the  remaining  angles. 

The  Precision.  —  From  Eq.  3  or  from  Eq.  5  it  is  evident 
that  each  of  the  angles  A,B,C  has  the  weight  2.  The  weight 
Pot  the  angle  —  A  -\-  B  is  given  by  (see  Art.  101) 


and     P=^ 


APPLICATION   TO   TRIANGULATION. 


335 


The  same  result  will  be  found  for  the  remaining  angles. 
Hence  after  the  station  adjustment  each  angle  has  the 
same  weight,  being  double  the  weight  of  a  single  measured 
angle  in  our  case  of  4  stations  sighted  at.  With  n  stations 

sighted  at  the  weight  of  an  adjusted  angle  would  be  ' 
times  the  weight  of  a  single  measured  angle. 

Ex.  i.  Given  at  station   Oswego,  in  the  triangulation  of  Lake   Ontario 
(Fig.  38), 

/n  =     80°      29'      46".  IO 

/13  =  107°        ig'        03". 28 

/14  =  I38°     12'     49"-44 

/as  ==     26°      49'      l6".6l 
4u=    57°     43'     oi".g6 
Au=    30°     53'     42".  88 
required  the  adjusted  values. 

Solution. 


46".  10 

63".  28 

109".  44 

218".  82 

i6".6i 

6i".g6 

78".  57      140".  25 

42".  88 

42".  88      i75"-94 

140".  25 

i75"-94 

218".  82 

iS6".35 

255"-83 

433"-  10 

46".  59 

63".  96 

108".  28 

218".  83     check. 

A 

B 

C 

and  the  adjusted  angles 


80° 

29' 

46" 

•59 

107° 

19' 

03* 

.96 

138° 

12' 

48" 

.28 

26° 

49' 

I?" 

37 

57° 

43' 

01", 

69 

30° 

53' 

44" 

32 

Check  Solution. 

Angle  A. 

Angle 

A'. 

Angle  C. 

46".  10       wt.  i 

3 

'.28       wt. 

i 

49".  44       wt.  i 

46 

'.67          '    \ 

2 

'•71 

" 

i 

46".  16        "    ^ 

.  47 

"•48         "    \ 

6 

"•56 

" 

i 

48".  06        "    ^ 

Mean,     46'. 59 


48". 28 


Similarly  for  the  remaining  angles. 

The  weight  of  each  adjusted  angle  is  twice  that  of  each  measured  angle. 

Ex.  2.  The  angles  at  station  Vanderlip  (Art.  142)  may  be  adjusted  in  the 
same  way  as  the  above. 


336  THE   ADJUSTMENT   OF   OBSERVATIONS. 

155.  The  General  Adjustment  (Angles).  —  The  angle 
and  side  equations  are  formed  as  explained  in  Arts.  117- 
139.  The  connection  between  the  local  and  general  adjust- 
ment is  through  the  weight  equations.  In  our  example  of 
4  stations,  in  which  every  angle  between  every  two  direc- 
tions is  measured,  the  weight  equations  are  in  three  groups, 
of  which  the  first  is  (see  Eq.  2,  p.  333) 


which  equations  are  moderately  simple  in  form. 

If  we  had  simply  closed  the  horizon  the  first   group  of 
weight  equations  would  have  been 


which  are  as  complicated  as  the  preceding. 

Hence,  in  a  net  in  which  the  angles  at  each  station  are 
measured  in  either  of  the  ways  indicated,  the  solution 
would  'be  simplified  so  far  as  the  local  adjustment  is  con- 
cerned, in  that  the  adjustment  at  each  station  is  of  a  fixed 
form,  but  in  the  general  adjustment  arising  from  the  angle 
and  side  equations  the  gain  is  comparatively  little. 

If,  however,  instead  of  finding  the  corrections  to  the 
angles  we  had  found  the  corrections  to  the  directions  of  the 
arms  of  the  angles,  the  weight  equations  become  much 
simplified,  and  therefore  also  the  whole  reduction.  This 
idea,  which,  I  believe,  is  due  to  Hansen,  will  now  be 
developed.* 

156.  The  Local  Adjustment  (Directions).  —  If  X, 
A,  B,  C  denote  the  readings  of  the  4  directions  i,  2,  3,  4 
from  O,  then  since  A  —  X,  B  —  X,  C  —  X  correspond  to  the 

*  See  Die  f>renss!schc  Landestriangulation.     Berlin,  1874,  scq. 


APPLICATION   TO   TRIANGULATION.  337 

A,  B,  C  in  the  angle  adjustment,  the  observation  equations 
may  be  written 

-X+A  -/„  =  *,. 

-X         +  B          -/,.  =  »„ 

-JT  +  c-4€  =  *I4 

-.*  +  /?        -/„  =  *'„  (0 

-A  +  C-/,t  =  ^t 

-JB  +  C-l.t  =  v,t 

and  the  normal  equations 

3x-  A-   B-   <:=-/„-/„-/„ 
-,r^-    5-    c=    /-/-/          2 


Adding  these  equations,  there  results 

0  =  0 

and  therefore  the  unknowns  cannot  be  found  without  some 
further  relation  connecting  them.  The  reason  of  the  inde- 
terminate form  is  that  directions  are  nothing  but  the  angles 
which  the  rays  Oi,  O2,  .  .  .  make  with  some  common  zero 
ray  whose  position  is  not  fixed,  and  which  may  therefore  be 
taken  arbitrarily.  To  carry  out  the  solution  it  will  be  most 
convenient  to  fix  the  zero  ray  by  making  the  arbitrary 
assumption 

X+A+B+C=o  (3) 

By  adding  this  to  each  of  the  normal  equations  they  re- 
duce to 

4'Y"  —      —    A   1    —    /I   3    ~    A    4 

*A--        /„_/„-/„ 

48=         /,,  +  /,,-/,,  (4) 

4C  /,4  +  /,4  +  '., 

which  give  the  values  of  X,  A,  B,  C  directly. 


338 


THE  ADJUSTMENT   OF   OBSERVATIONS. 


The  computation  may  be  rendered  quite  mechanical  by 
arranging  in  tabular  form  : 


I 

2 

3 

4 

Sums. 

4, 

4s 

4, 

Sum, 

4s 

4, 

Sum2 

4  4 

Sum3 

-  Sum, 

-  Sum, 

-  Sum3 

4X 

4A 

AB 

4C 

X 

A 

B 

C 

Check 

The  transformation  of  the  normal  equations  (2)  into  (4) 
by  means  of  the  arbitrary  relation  (3)  is  allowable.  For 
since  the  sum  of  the  coefficients  of  the  unknowns  in  each  of 
equations  2  is  zero,  whatever  values  of  X,  A,  B,  C  satisfy 
those  equations  X-\-a,  A  -\-a,  B-\-a,  C  -\-  a,  where  a  is  any 
constant,  will  also  satisfy  them.  Hence  whatever  set  of 
values  is  taken  to  satisfy  the  equations,  the  differences 
A—  X,  B  —  X,  C—X  will  be  the  same  in  value.  There- 
fore by  arbitrarily  fixing  the  zero  direction  we  find  deter- 
minate values  for  the  corrections  to  the  other  directions. 


Ex.  Take  that  of  Ex.  i,  Art.  154. 
seconds  only,  is 


The  tabular  scheme,  writing  down  the 


46".  i  o 

63".  28 

109".  44 

218".  82 

i6".6i 

6i".g6 

78"-57 

42  ".88 

42".  88 

—  218".  82 

-  78".  57 

—  42".  88 

—  218".  82 

-  32".  47 

37".  01 

214".  28 

-  54"  -70 

—  8".  12 

9".  25 

53"-  57 

X 

A 

B 

C 

and  A  —  X,  B  —  X,  C—X  give  the  same  values  of  the  angles  as  found  before. 
Hence,  whether  we  find  the  local  corrections  to  the  angles  or  to  the  directions 


APPLICATION   TO   TRIANGULATION. 


339 


of  the  arms  of  the  angles,  the  result  is  the  same.     One  method  or  the  other 
may,  therefore,  be  used,  as  is  most  convenient. 

To  avoid  the  use  of  large  numbers  certain  approximate  values  may  be 
assumed  for  the  directions,  and  the  corrections  to  these  approximate  values 
found.  (Compare  Art.  Si.)  Thus  if  (X),  (A),  (B),  (C)  denote  the  corrections 
to  assumed  values  of  X,  A,  B,  C,  we  may  proceed  as  follows : 

Assumed  approximate  angles 

Sodus,             o°  oo'  oo*  +  (A') 

Vanderlip,    80°  29'  46"  +  (A) 

Duck,           107°  19'  03"  +  (B) 

Stony  Pt.,    138°  12'  4 

Tabular  Form. 


I 

2 

3 

4 

Sums. 

+  o".io 

+  0".2S 

+  o'.44 

+  0".S2 

-  o"-39 

-  i  ".04 

-  i  -43 

—  3".  12 

—  3'.  12 

—  0".S2 

+  i"-43 

+  3"-  12 

—  0".S2 

+  i  "-53 

+  3".  oi 

—  3"-?6 

—  o"  .  20 

+  o".38 

+  °"-75 

—  o".g4 

(X) 

(A) 

(S) 

(0 

and  the  adjusted  angles  are  as  before. 

157.  The  General  Adjustment  (Directions). — The 

form  of  the  normal  equations  4,  Art.  156,  shows  that  at 
each  station  the  quantities  X,  A,  .  .  .  are  determined  inde- 
pendently of  one  another.  The  weight  of  each  direction 
at  a  station,  as  shown  by  the  weight  equations,  is  repre- 
sented by  the  number  of  stations  sighted  at.  If,  therefore, 
(i),  (2),  .  .  .  denote  the  corrections  to  the  values  A',  A,  .  .  . 
locally  adjusted,  which  arise  from  the  angle  and  side  equa- 
tions, we  may  consider  X,  A,  ...  as  independent  quanti- 
ties, all  of  equal  weight  and  subject  to  certain  rigorous  con- 
ditions, and  proceed  to  carry  out  the  solution  according  to 
the  simple  form  of  Art.  no.  Hence,  as  the  solution  breaks 
into  two  simple  problems  of  adjusting  quantities  as  if  inde- 
pendently observed,  we  see  the  advantage  of  adjusting  the 
directions  instead  of  the  angles. 

44 


340 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


'  Ex.  The  quadrilateral   Buchanan,   Brule,  Aminicon,   Lester,  in  the   tri- 
angulation  of  Lake  Superior. 


Direction. 

At  Buchanan,     2-1 


At  Brule, 


3-2 


At  Aminicon,     8-7 


At  Lester, 


Measured  Angle.  Local  Adj .  Angle. 

47°     57'     36". 25  36".  21 

97°     26'     41". 29  4i"-32 

49°     29'     05".  15  5".  1 1 


5-4 

40° 

25' 

47"-49 

47"-36 

6-4 

71° 

58' 

27".  31 

27".  45 

6-5 

31° 

32' 

40".  22 

40".  09 

8-7 

37° 

47' 

05".  51 

5"-  1? 

9-7 

97° 

51' 

03".  1  1 

3"-45 

9-8 

60° 

03' 

58".  62 

58".  28 

II-IO 

5i° 

oo' 

39"-  77 

40".  oo 

12-10 

92° 

43' 

49".  42 

49".  20 

1  2-1  1 

41° 

43' 

o8".97 

9".  2O 

The  local  adjustment  is  carried  through  as  in  Art.  154,  it  being  more  con- 
venient to  adjust  the  angles  directly  father  than  the  directions. 

Taking  the  locally  adjusted  angles  as  independent,  we  next  find  the  cor- 
rections to  the  directions  arising  from  the  angle  and  side  equations. 

The  angle  equations,  formed  in  the  usual  way  from  the  triangles  Buchanan, 
Brule,  Aminicon  (e  =  i"-37)  ;  Brule,  Aminicon,  Lester  (e  =  i".i9);  Lester, 
Buchanan,  Brule  (s  =  i".  19) ;  and  the  side  equation,  from  the  quadrilateral 
itself  (pole  at  Lester),  are 

-(i) +  (2) -(4) +  (6)-  (8)+  (9)=-o'.57 
-  (i)  +  (3)  -  (5)  +  (6)  -  (10)  +  (ii)  =  -  o".22 
-(4) +  (5) -(7) +  (9) -(«)  +  (")  =  +  i".i8 

-O.I4(l) -0.90(2)+  1.04(3)  +  I- 24 (4) -2. 95(5) 

+  i .  72(6)  +  i .  50(7)  -  i .  36(8)  -  o.  14(9)  =  -  o" .  80 

The  number  of  stations  pointed  at  from'  each  station  occupied  being  3,  the 
weight  of  each  locally  adjusted  angle  is  the  same  throughout  the  net. 
Hence  we  have  the  correlate  normal  equations 

I.  II.  III.  IV. 

+  6.0         +2.0        +2.0         +    0.94        =—0.57 

+  2.O  +  6.O  —  2.O  +      5.84  =  —  O.22 

+  2.0        —2.0         +6.0          -    5.84        =  +  1.18 
+  0.94      +5.84      —5.84       +19.22       =— 0.80 


I.  =  —  0.273 
II.  =  +0.136 


III.  =  +0.377 

IV.  =  +  0.045 


APPLICATION   TO   TRIANGULATION.  341 

and  the  corrections 

C1)  —  +o".  13  (7)  =  — o".3i 

(2)=— 0".3I  (8)=r+o".2I 

(3)=s+o".i8  (9)=+o".io 

(4)  =  —  o'.os  (10)  =—  o".i4 

(5)=+o'.n  (ii)  =  — o".24 

(6)=—  o".o6  (i2)=:  +  o".3S 


On  the  Breaking  of  a  Net  of  Triangulation  into  Sections  for 
Convenience  of  Solution. 

158.  In  a  long  chain  of  triangulation  or  in  a  complicated 
net  the  simultaneous  solution  of  the  condition  equations 
would  be  very  troublesome,  not  from  any  principle  in- 
volved, but  from  its  very  umvieldiness.  Accordingly  it  is 
necessary  to  break  the  work  into  sections  and  solve  each 
section  by  itself.  As  this  breaking  into  sections  causes 
more  or  less  disturbance  of  the  local  conditions  at  the  lines 
of  breaking  off,  each  section  should  be  as  large  as  can  be 
conveniently  managed,  and  the  lines  of  breaking  off  should 
be  so  chosen  as  to  disturb  as  few  conditions  as  possible. 
By  the  method  of  adjustment  explained  in  Arts.  153-157 
much  larger  sections  can  be  taken  than  by  any  other. 
This  is  a  strong  argument  in  favor  of  its  use. 

The  contradictions  that  occur  at  the  lines  of  breaking  off 
of  the  several  sections  are  most  conveniently  bridged  over 
by  means  of  the  principle  of  Art.  115.  As  an  example  we 
may  cite  the  reduction  of  the  principal  triangulation  of  the 
British  Ordnance  Survey.  There  were  920  condition  equa- 
tions to  be  satisfied  in  the  net.  The  following  was  the 
method  of  solution  employed:* 

"  The  triangulation  was  divided  into  a  number  of  parts 
or  figures,  each  affording  a  not  unmanageable  number  of 
equations  of  condition.  One  of  these  being  corrected  or 
computed  independently  of  all  the  rest,  the  corrections  so 
obtained  were  substituted  (so  far  as  the}7  entered)  in  the 

*  Account  of  the  Principal  Triangulation,  p.  272. 


342  THE   ADJUSTMENT   OF   OBSERVATIONS. 

equations  of  condition  of  the  next  figure,  and  the  sum  of 
the  squares  of  the  remaining  corrections  in  that  figure  made 
a  minimum.  The  corrections  thus  obtained  for  the  second 
figure  were  substituted  in  the  third,  and  so  on." 

In  the  triangulation  of  Mecklenburg*  this  method  was 
carried  out  even  more  systematically.  The  adjustment 
was  divided  into  five  groups  of  22,  22,  22,  21,  22  condition 
equations  respectively.  The  corrections  resulting  from  the 
solution  of  group  1.  were  carried,  so  far  as  they  entered,  into 
group  II.,  and  this  group  solved,  and  so  on  through  groups 
III.,  IV.,  V.  The  whole  operation  was  repealed  four 
times,  and  the  small  contradictions  still  remaining  were 
distributed  empirically. 

For  an  interesting  conference  on  the  whole  question 
see  Comptes  Rendus  de  r Association  Gcode'sique  Internationale, 
1877. 

Adjustment  of  a   Triangulation  for  Closure  of  Circuit. 

159.  In  the  adjustment  of  a  triangulation  we  have  so  far 
considered  it  only  with  reference  to  a  single  measured  base. 
We  have  seen  how  at  each  station  the  discrepancy  arising 
from  sum  angles  and  from  closure  of  the  horizon  can  be  got 
rid  of,  and  also  how  in  a  net  joining  several  stations  the 
conditions  arising  from  closure  of  triangles  and  from 
equality  of  lengths  of  sides  computed  by  different  routes 
can  be  satisfied.  There  remains  the  question  as  to  the 
mode  of  procedure  when  several  bases  enter  whose  lengths 
are  known  and  whose  positions  have  been  fixed  astronomi- 
cally. Special  cases  would  be  where  a  circuit  of  triangula- 
tion closed  on  the  initial  line,  and  where  a  secondary  system 
is  to  be  made  to  conform  to  two  lines  in  a  primary  system, 
the  primary  lines  being  assumed  to  be  known  in  length  and 
position. 

The  conditions  to  be  satisfied  in  the  adjustment  are  four 
in  number-— that  the  value  of  a  base  computed  from  another 

*  Grosshcrzoglich  mecklcnburgi&che  Landesi'i:ri>iessung.     Schwerin,  1882. 


APPLICATION   TO   TRIANGULATION.  343 

should  agree  with  the  measured  value  in  azimuth,  in  length, 
and  in  latitude  and  longitude  of  one  of  the  end  points. 

The  measured  angles  of  the  triangles  connecting  the 
bases  having  been  already  adjusted  with  reference  to  one 
base  for  the  local  and  general  conditions,  the  additional  cor- 
rections necessary  to  satisfy  the  closure  of  the  circuit  will 
be  small.  But  little  difference  in  the  results  will  therefore 
be  found  by  making  a  simultaneous  solution  of  all  of  the 
condition  equations  of  closure  and  a  solution  by  successive 
approximation  according  to  the  method  of  Art.  115. 

We  shall  consider  only  a  single  chain  of  triangles,  all  tie 
lines  of  the  system  being  rejected.  This  on  account  of  sim- 
plicity, and  also  for  the  reason  stated  above,  that,  in  good 
work  the  corrections  being  small,  it  gives  results  practically 
close  enough  —  nearly  the  same,  in  fact,  as  a  rigorous  solu- 
tion. If  thought  necessary  in  any  special  case  a  rigorous 
solution  of  the  condition  equations  can  be  carried  out  by 
the  method  of  correlates. 

1 60.  Adjustment  for  Discrepancy  in  Azimuth.— 
Let  1-2  and  5-6  be  two  bases  connected  through  inter- 
mediate stations  3,  4  by  a  single  chain 
of  the  best-shaped  triangles  that  can 
be  selected  from  the  net. 

In  computing  the  base  5-6  from  1-2 
the  sides  1-4,  3-4,  3-6  are  at  once  sides 
of  continuation  and  bases,  according 
to  the  triangles  considered.  For  example,  in  the  triangle 
i  2  4  the  side  1-4  is  a  side  of  continuation  from  the  base  1-2, 
but  in  134  the  side  1-4  is  a  base  with  reference  to  3-4  as  a 
side  of  continuation. 

In  the  chain  of  triangles  let 

Alt  A^  ...  be  the  angles  opposite  to  the  sides  of  con- 
tinuation. 

/?,,  B^,  .  .  .  the  angles  opposite  to  the  bases  in  order  of 
computation. 

C}}  £*„,...   the  angles  opposite  to  the  flank  sides. 


344  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Let  z,  zl  denote  the  values  of  the  measured  azimuth  of  the 
bas.es  1-2,  5-6  respectively.  These  values  are  assumed  to 
be  correct,  and  receive  no  change  in  the  adjustment. 

A  geodetic  computation*  of  the  azimuth  of  the  base  5-6 
from  the  base  1-2  is  now  made,  using  the  values  of  the 
angles  of  the  intervening  triangles  resulting  from  the  adjust- 
ment for  local  and  general  conditions.  Call  the  value  of 
the  azimuth  of  5-6  computed  in  this  way  z' . 

Now,  reckoning  azimuth  in  the  usual  way  from  the 
south  as  origin,  and  the  direction  of  increase  from  south  to 
west,  it  is  easily  seen,  by  passing  from  1-2  along  the  sides 
1-4,  4-3,  3-6,  that  the  excess  4  of  the  observed  value  zl  over 
the  computed  value  z'  of  5-6  is  given  by 

-(Q  +  (Q-(Q  +  (Q  =  4  (i) 

where  (£7,),  (Cy),  .  .  .  denote  the  corrections  to  the  angles 
£*„  £,,...  This  is  the  azimuth  condition  equation. 

In  order  that  the  correction  arising  from  the  azimuth 
equation  may  not  disturb  the  conditions  of  closure  existing 
among  the  angles  of  the  triangles,  it  is  necessary  that  the 
corrections  to  the  angles  should  satisfy  the  conditions 


The  unknowns  in  equations    I,    2    are    subject    to    the    re- 
lation 

'  -  .  .  =  a  rain. 


Call  £,,  k^  k^  /£4  the  correlates  of  equations  (2),  and  k  the  cor 
relate  of  equation  (i),  and  we  have  the  correlate  equations 


*.     = 

\  —  k  —  (Q       k,  +  k  =  (Q       k.  -  k  =  (Q       £4  +  ^  =  (Q 


*  For  methods  of  doing  this  see  Lee's  Tables  and  Formulas,  Washington,  1873  ;  Coast  Sur- 
vey Report,  1875. 


APPLICATION   TO   TRIANGULATION.  345 

whence  the  normal  equations 

3^  k  =  o 

3^3  -f   k  =  o 


If  we  had  ;/  triangles  instead  of  four,  the  normal  equations 
would  be  of  similar  form,  and,  solving,  we  should  find 


*= 

2n 


and  therefore  the  corrections 


~ 


Hence  the  rule  :  Divide  the  excess  of  the  observed  over  the 
computed  azimuth  by  the  number  of  triangles,  and  apply  one-half 
of  this  quantity  to  each  of  the  angles  adjacent  to  the  flank  sides 
on  one  side  of  the  chain,  and  the  total  quantity,  with  the  sign 
changed,  to  the  third  angle.  The  signs  are  reversed  for  the 
angles  on  the  other  flank. 

If  the  azimuth  mark  is  not  on  a  triangulation  side  the 
line  of  azimuth  may  be  swung  on  to  such  a  side  by  adding 
the  angle  between  the  mark  and  the  side. 


34^ 


THE  ADJUSTMENT  OF  OBSERVATIONS. 


Ex.  The  sketch  represents  the  secondary  triangulation  of  Long  Island 
Sound.     The  sides  1-2,  14-15  are  the  lines  of  junction  with  the  primary  sys- 


tem.    These  lines  are  assumed  to  remain  unchanged  in  azimuth  in  the  adjust- 
ment, and  the  secondary  system  is  to  be  made  to  conform  to  them. 

The  main  chain  of  triangles  joining  the  two  primary  lines  is  indicated  in 
the  figure  by  heavy  lines.  The  number  of  these  triangles  is  n.  The  system 
has  been  adjusted  for  local  and  geometrical  conditions,  and  the  resulting 
angles  of  these  triangles  are  as  follows  : 


Angle. 

log  sin 
diff.  i" 

Sph. 
excess. 

Angle. 

log  sin 
diff.  i" 

Sph. 
excess. 

A, 

82°  49'  19".  25 

0.27 

A, 

58°    25'   08".  08 

1.29 

B, 

64°  55'  40".  32 

0.99 

3"-66 

B-, 

82°  16'  i6".i6 

0.29 

3"-63 

c, 

32°  15'  04".  09 

C-, 

39°  18'  39"-39 

A* 

52°  37'  47"-98 

1.61 

A& 

69°  12'  06".  02 

0.80 

B* 

74°  10'  50".  56 

0.60 

4".  96 

£* 

71°  58'  54"-i8 

0.68 

3"-°3 

c, 

53°  n'  26".  42 

cs 

38°  49'  02".83 

A3 

69°  20'  1  6".  03 

0.79 

A, 

59°  18'  26".  33 

1.25 

B* 

67°  43'  43"-  33 

0.86 

3".  53 

B§ 

102°  08'  26".  94 

-0.45 

i  "-33 

C3 

42°  56'  04".  16 

ca 

18°  33'  08".  05 

A, 

39°  31'  29".  66 

2-55 

A^0 

67°  38'  52".  77 

0.87 

B, 

120°  36'  36".  1  1 

-1-25 

i"-3i 

•Bio 

70°  26'  23".  26 

0.75 

2".  41 

c< 

19°  5i'  55"-54 

Go 

41°  54'  46".  38 

A, 

83°  19'  3i"-o8 

0.25 

An 

29°  43'  55"-  99 

3.69 

B& 

68°  57'  47"-  75 

o.Si 

i  ".42 

£n 

55°  23'  09".  36 

1-45 

2".  1  3 

c, 

27°  42'  42".  58 

CM 

94°  52'  56".  78 

At 

87'  25'  26".  66 

0.09 

B, 

44°  10'  45".  25 

2.17 

3".  47 

Ct 

48°  23'  5  1  ".56 

APPLICATION   TO   TRIANGULATION.  347 

A  geodetic  computation  for  latitude,  longitude,  and 
azimuth  was  carried  through  from  the  line  1-2  to  14-15, 
using  the  above  angles.  It  was  found  that  approximately 

observed  az.  of  14-15  —  computed  az.  of  do.  =  —  2". 93 

Hence  the  corrections  to  the  angles  of  the  triangles  for  this 
discrepancy  in  azimuth  are  for  the 

first  triangle  (/!,)  —  —  o".  13     second  triangle  (A^)  =  -\-o".  13 

(/>',)  =-o".  1 3  W  =  +  o*.is 

(Q  =  -f  0".26  (Q  =  -  o".26 

and  so  on. 

161.  Adjustment  for  Discrepancy  in  Bases.— This 
is  fully  explained  in  Arts.  168-170. 

Using  the  last  form  given  in  Art.  170,  we  may  write  the 
base-line  equation 

\dA(A)-dB(B)-\  =  l 

from  which,  using  the  values  of  A^  />,,  .  .  .  found  in  the 
azimuth  adjustment  as  first  approximations,  further  correc- 
tions to  the  angles  are  found,  as  in  equations  5,  Art.  170. 
Since  the  angles  C  do  not  enter  into  this  adjustment,  the 
corrections  resulting  will  not  disturb  the  adjustment  for 
azimuth  already  made. 

The  advantage  of  this  method  of  proceeding  is  that  the 
subsequent  work  does  not  disturb  any  adjustment  already 
made  and  thus  render  it  necessary  to  make  a  new  approxi- 
mation. The  labor  is  reduced  to  a  minimum,  and  the  re- 
sults obtained  are  practically  close  enough. 

For  an  example  see  Ex.  i,  Art.  170. 

162.  Adjustment    for    Discrepancy    in    Latitude 
and  Longitude. — The  corrections  to   the   angles  arising 
from   discrepancy  in  azimuth  and  in  bases  having  been  ap- 
plied, the  value  of  one  base   computed   from  another  will 
a<;ree   with   the   measured   value   in   direction  and    length. 

o  *— 

45 


348  THE   ADJUSTMENT   OF   OBSERVATIONS. 

The  discrepancy  in  position,  as  shown  by  the  differences  be- 
tween observed  and  computed  latitudes  and  longitudes, 
alone  remains.  This  discrepancy,  being  small,  may  be 
eliminated  closely  enough  by  distributing  it  proportionally 
from  one  end  of  the  chain  of  triangles  to  the  other,  accord- 
ing to  Bowditch's  rule  as  given  in  Ex.  5,  Art.  no— that  is, 
the  error  in  latitude  in  proportion  to  the  longitudes,  and 
the  error  in  longitude  in  proportion  to  the  latitudes  of  the 
several  stations.  Each  station,  being  thus  made  slightly 
eccentric,  is  next  reduced  to  centre,  when  the  whole  net 
will  be  consistent. 


CHAPTER  VII. 

APPLICATION   TO   BASE-LINE    MEASUREMENTS. 

163.  During  the  present  century  two  forms  of  apparatus 
have  been  used  in  the  measurement  of  primary  bases,  the 
compensation  bars  and  the  metallic-thermometer  appa- 
ratus. On  the  English  Ordnance  Survey  the  two  principal 
lines,  the  Lough  Foyle  and  Salisbury  Plain  bases,  were 
measured  with  the  Colby  compensation  bars.  Most  of  the 
bases  of  the  U.  S.  Coast  Survey  and  five  of  the  eight  bases 
of  the  U.  S.  Lake  Survey  were  measured  with  the  Bache- 
Wiirdemann  compensation  apparatus.  On  the  Continent 
of  Europe  the  Bessel  metallic-thermometer  apparatus  is 
very  generally  used. 

Indications  are  not  wanting  that  both  forms  will  be  sup- 
planted before  long  by  an  apparatus  consisting  of  simply  a 
single  metallic  bar.* 

The  essential  part  of  any  form  of  base  apparatus  consists 
of  one  or  two  bars  of  metal,  usually  from  4  to  6  metres  in 
length  and  of  about  40  X  15  mm.  cross-section.  If  the  ex- 
treme points  of  a  bar  are  the  limits  of  measure,  so  that  a 
measurement  is  made  by  contact  [end-measures],  two  bars 
are  necessary.  If,  however,  the  length  of  a  bar  is  con- 
sidered to  lie  between  two  marks  made  on  it  [line-meas- 
ures], so  that  in  a  measurement  the  transition  from  one  bar 
to  the  next  depends,  not  on  the  stability  of  the  bar,  but  on 
some  outside  appliance,  only  one  bar  is  necessary. 

Descriptions  of  the  various  forms  of  apparatus  used 
on  different  surveys  will  be  found  in  reports  of  those 
surveys. 

*  See  Ibanez,  Zfitschr.  fiir  Instrumentenkiinrif,  1881.     Also  Art.  166. 


350  THE   ADJUSTMENT   OF   OBSERVATIONS. 

164.  Precision  of  a  Base-Line  Measurement.— For 

clearness  it  will  be  necessary  to  outline  the  principles  on 
which  the  measurement  is  made. 

First,  we  must  find  the  length  of  the  measuring  bar  in 
terms  of  some  standard  of  length  ;  and  as  the  measurements 
of  the  line  itself  are  made  at  various  temperatures,  the  co- 
efficients of  expansion  of  the  metals  in  the  measuring  appa- 
ratus must  also  be  known.  Comparisons  must,  therefore, 
be  made  with  the  standard  during  wide  ranges  of  tempera- 
ture;  and  as  these  comparisons  are  fallible,  the  results 
found  for  length  and  expansion  will  be  more  or  less  er- 
roneous. 

The  principle  involved  in  the  measurement  is  exactly 
the  same  as  in  common  chaining  with  chain  and  pins. 
There  are,  indeed,  various  contrivances  for  getting  a  pre- 
cision not  looked  for  in  chaining,  such  as  for  aligning  the 
measuring  bar,  for  finding  the  inclination  of  each  position 
of  the  bar,  and  for  establishing  fixed  points  for  stopping  at 
and  starting  from  in  measurement.  But  these  make  no 
change  in  the  essential  principle. 

The  errors  in  the  value  of  a  base  line  may,  therefore,  be 
considered  to  arise  from  two  principal  sources,  comparisons 
and  measurement.  Experience  has  shown  that  the  main 
error  arises  from  the  comparisons,  and  that,  even  it  our 
modes  of  measuring  the  base  line  itself  were  perfect,  the 
precision  of  the  final  value  would  be  but  little  increased  so 
long  as  the  methods  of  comparison  are  in  their  present 
state.  Thus  in  the  Lake  Survey  primary  bases,  if  the  field 
work  had  been  without  error,  the  total  p.  e.  of  the  bases 
would  have  been  diminished  only  about  ^  part. 

These  errors  differ  essentially  in  character.  An  error 
arising  from  the  comparisons,  being  the  same  for  each  bar 
measurement,  is  cumulative  for  the  whole  base,  while  errors 
arising  in  the  measurement  of  the  base  itself,  were  the 
measurements  repeated  often  enough  and  the  conditions 
sufficiently  varied,  would  tend  to  mutually  balance,  and 
could,  therefore,  be  treated  by  the  strict  principles  of  least 


APPLICATION   TO   BASE-LINE    MEASUREMENTS.  351 

squares.  But  as  the  number  of  measurements  is  not  often 
more  than  2  or  3,  and  as  these  are  made  usually  at  about 
the  same  season  of  the  year,  only  a  comparatively  rough 
estimate  of  the  precision  is  to  be  looked  for. 

As  a  check  on  the  field  work  a  base  is  usually  divided 
into  sections  by  setting  stones  firmly  in  the  ground  at 
approximately  equal  intervals  along  the  line,  so  that  in- 
stead of  being  able  to  compare  results  at  the  end  points 
only,  \ve  may  compare  results  just  as  well  at  6  or  8  points. 
In  this  way  a  better  idea  of  the  precision  of  the  work  is 
obtained,  as  we  have  6  or  8  short  bases  to  deal  with  instead 
of  a  single  long  one. 

We  proceed  now  with  the  problem  of  determining  the 
precision  of  measurement.  Ft  may  be  stated  as  follows: 
A  base  is  measured  in  «  sections  with  a  bar  of  a  certain 
length,  each  section  being  measured  ;/,  times.  By  the  first 
measurement  the  first  section  contains  J//  bars,  the  second 
M"  bars,  .  .  .  ;  by  the  second  measurement  the  first  sec- 
tion contains  MJ  bars,  the  second  Mf  bars,  .  .  .  ;  and  so 
on.  The  weights  of  the  measurements  in  order  being 
A'»  A'»  •  •  •  /  P"  •>  A">  •  •  •  /  •  •  •  respectively,  required  the 
m.  s.  e.  of  the  most  probable  value  of  the  base. 

Let          F,  —  most  prob.  value  of  first  section 

Fa  =  most  prob.  value  of  second  section 

then  we  have  the  observation  equations 
First  section,  V,  —  M{  •  =  ?>/      wt.  />,' 

V,-M{'=,v;'    wt.  A" 

Second  section,        F,  —  J//  :  =  r>/      wt.  // 
V,  -  M,"  =  v,"     wt.  A" 

and  so  on. 

Now,  either  of  two  assumptions  may  be  made. 


352  THE   ADJUSTMENT   OF   OBSERVATIONS. 

(a)  In  the  first  place,  that  the  precision  of  the  measure- 
ment of  each  bar  is  the  same  throughout  the  different 
sections. 

We  have,  then,  nnl  equations  containing  n  unknowns, 
and  the  normal  equations  are 


whence    F,,  F2,  .   .  .  are  known,  and    therefore    the    whole 
line  V=  V,  -f  Fa  +  .  .  .  -f  Vn  is  known. 

The  mean-square  error  /j.  of  an  observation  of  weight 
unity  —  that  is,  of  a  single  measurement  of  a  bar  —  is  given 
by  (see  Art.  99) 


[fivv] 


No.  of  obs.  —  No.   of  indep.  unknowns. 


Now,  the  length  of  the  measuring  bar  being  taken  as  the 
unit  of  measurement,  the  weight  of  a  section,  as  depending 
on  the  measurement,  may  be  expressed  in  terms  of  the 
number  of  bars  measured. 

For  since  ft  is  the  m.  s.  e.  of  a  measurement  of  a  single 
bar,  the  'm.  s.  e.  of  the  measurement  "of  a  length  of  M  bars 

is  n  VM.     Hence    =^  is   the    weight   of  a   measurement   of 

length  M  when  the  weight  of  a  measurement  of  the  unit  of 
length  is  unity. 

Writing,   therefore,   for  the    weights  /  their  values   in 
terms  of  M, 


/         I          Vvi 

y  «(«,-  i)\Jf 


vv~\ 
M\ 


In  the  case  usually  occurring  in  practice,  where  the  line 
is  measured  twice,  we  may  put  this  formula  in  a  form  more 


APPLICATION   TO   BASE-LINE    MEASUREMENTS. 


353 


convenient  for  computation.  For  if  the  first  measurement 
of  the  w,  sections  gave  lengths  Mlt  Mt,  .  .  .,  and  the  second 
measurements  gave  lengths  J/,-)-^,  M^-\-dv  .  .  .  for  the 
same  sections  in  order,  then,  since 


we  have  for  the  m.  s.  e.  of  one  measurement  of  a  bar  and  for 
the  mean  of  two  measurements  respectively 


Hence  the  m.  s.  e.  of  a  single  measurement  and  of  the  mean 
of  the  two  values  of  the  whole  base  are 


the  number  of  bars  in  the  line  being  [M]. 


Ex.  The  Bonn  Base,  measured  in  1847,  near  Bonn,  Germany,  with  the 
original  Bessel  metallic-thermometer  apparatus.  The  base  was  a  broken 
one,  the  two  parts  making  an  angle  of  179°  23'.  Each  part  was  measured 
twice,  as  follows:* 


Differences. 

No.  of  bars. 

Northern  Part,  Sec.  i 

L 
—  0.183 

116 

Sec.  2 

+  o.cn)4 

87 

Sec.  3 

—  0.013 

6  1 
-  264 

Southern  Part,  Sec.  i 

—  0.007 

92 

Sec.  2 

+  o.(xj5 

60 

Sec.  3 

+  0.757 

131 

283 

*  Das  rhcinischf  Drciecksnct:.     Berlin,  1876. 


354  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Hence  the  m.  s.  e.  of  the  northern  part,  arising  from  errors  of  measurement 
only,  is 


y_     .094' 

6    ""    87 
and  the  m.  s.  e.  of  the  southern  part  is 


L 
--     =  ±  0.093 


283/.0072       .cos'2       -757'2 

-+-    -  +  -  L 

92  60  131 


\  L 

=±0- 

/ 


The  other  two  main  sources  of  error  are  : 

1.  Error  in  comparison  of  the  measuring  bars  with  one  another. 

2.  Error  in  the  determination  of  their  length. 

The  m.  s.  e.  arising  from  these  sources  are  respectively 

L  L 

±  0.386,    ±  0.313  for  the  northern  part 

±  0.391,    ±  0.335  f°r  the  southern  part 
Remembering  that  these  latter  errors  are  systematic,  we  have,  finally, 


m.  s.  e.  of  base  =  Kogs^  .327'^+  (.386  +  .391)'-  +  (.313  +  .335)- 

L 

=  1.07 

(b)  In  the  second  place,  if  we  assume  that  the  law  of 
precision  of  the  measurements  of  the  different  sections  is 
unknown,  and  that  these  sections  are  independent,  we  have 
for  the  mean  of  the  values  of  the  several  sections  and  their 
m.  s.  e. 


v  _ 

J&wL 


since  p,     ,,=  m      .  i 


v  _ 


n     2  -  a  a    .  ,          .  „ 

/  K    —  r  .  -,/  —      —  v  -  —  r  Since  A  =ff=.  .  .  = 

[AJ<X  -  i)      «,(«,  -  i) 


APPLICATION   TO    BASE-LINE    MEASUREMENTS.  355 

If  V  denotes  the  whole  line,  so  that 

r=*r,+   *r.+   .     •     •    +Vn 

then,  since  the  measurements  are  independent, 


and  the  (m.  s.  e.)2  of  a  single  measurement  of  the  line 


The  number  of  bars  in  the  line  being  [M],  we  have  for  the 
average  value  of  the  (m.  s.  e.)2  of  a  single  measurement  of  -a 
bar 


If,  for  example,  the  line  has  been  measured  twice,  and 
d^  */„,...  ^4  are  the  differences  of  the  measurements  ot  the 
several  sections,  then 


and  therefore 


r    »  2  i 

and  the  (in.  s.  e.)2  of  a  single  measurement  of  a  bar  is  i  FTT 


46 


356 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


Ex.  The  Chicago  Base,  measured  in  1877  with  the  Repsold  metallic- 
thermometer  apparatus  belonging  to  the  U.  S.  Engineers.  The  base  was 
divided  into  8  sections  and  was  measured  twice. 


Section. 

No.  of  Bars. 

Diff.  of  Measures. 

I. 

227.25 

mm 
-  1-3 

II. 

230.25 

+  2.5 

III. 

234.50 

+  2.3 

IV. 

232. 

+  0.7 

V. 

231. 

+  i.5 

VI. 

225. 

+  i.i 

VII. 

300.50 

+  1.3 

VIII. 

196.80 

—  O.2 

Taking  the  errors   of  the   different   sections  as  independent,  the  p.  e.  of  the 
mean  of  the  two  measures  of  the  base  is 


:.46 


0.6745  \ 

4 


The  p.  e.  arising  from  the  other  sources  of  error  were 

mm 

(1)  measuring  bar,  ±  6.38 

(2)  metallic  thermometer,  ±  2.82 

(3)  elevation  above  mean  tide,  N.  Y.,    ±  0.36 

Assuming  these  to  be  independent,  the  p.  e.  of  the  Chicago  Base  at  sea  level  is 

mm 

Vi.46'2  +  6.38-  +  2.82s  +  0.36-  =  7.14 

Of  the  two  assumptions  (a)  and  (b),  the  first  is  due  to 
Bessel  and  the  second  to  Andrse.  For  a  more  elaborate 
discussion  of  the  points  involved  see  Astron.  Nac/ir.,  Nos. 
1924,  1935,  and  V.  J.  Sclir.  d.  Astr.  Ges.,  1878,  p.  69. 

165.  Length  of  Base  and  Number  of  Bases  neces- 
sary in  a  Triangulation. — In  computing  a  side  of  a 
triangle  from  another  and  shorter  side  as  base,  the  loss  of 
precision  arising  from  the  influence  of  the  acute  angle 
opposite  to  the  base  is  the  greater  the  more  acute  that 
angle  is.  To  guard  against  this,  if  a  measured  base  were  a 
side  of  a  triangle  it  would  be  necessary  that  its  length  be 


APPLICATION   TO   BASE-LINE   MEASUREMENTS. 

about  the  average  length  of  the  triangle  sides  of  the  sys- 
tem. In  the  older  work  the  practice  was  to  measure  long 
bases,  even  as  long  as  20  kilometres  and  over.  The  modern 
practice  was  introduced  by  Prof.  Schwerd,  of  Speyer,  Ger- 
many, in  1819.  By  measuring  a  short  base  of  860**  and 
checking  on  to  a  line  computed  from  a  base  of  15  kilo- 
metres, he  found  a  result  agreeing  within  om. i.  His  con- 
elusion  was  that  "  with  a  small  expenditure  of  time,  trouble, 
and  expense  the  base  of  a  large  triangulation  can  be  de- 
termined by  a  small  exactly  measured  line." 

A  similar  conclusion  was  reached  by  Gen.  Ibanez  as  the 
result  of  measuring  the  base  of  Madridejos,  Spain,  in  1859. 
He  says:  "  The  question,  so  much  disputed  between  French 
and  German  geometers,  as  to  whether  it  is  necessary  to 
measure  long  bases,  or  if  short  bases  are  sufficient,  had 
occupied  the  attention  of  the  observers.  Taking  advantage, 
therefore,  of  the  favorable  opportunity  which  presented 
itself  to  them,  they  proposed  to  compare  the  results  ob- 
tained from  the  direct  measure  of  the  whole  base  with  those 
calculated  from  a  special  triangulation  depending  on  the 
central  section  of  the  base."f 

The  adjustment  of  this  triangulation  net  gave  rise  to  36 
angle  equations  and  28  side  equations  containing  90  unknown 
quantities.  The  net  is  shown  in  the  figure. 

"  We  give  in  the  following  table  the  results  of  the  direct 
measures  reduced  to  the  sea  level,  and  their  comparison 
with  the  values  found  trigonometrically  : 

Measured.  Triangulation. 

Fi9-47 

3077459  3077-462 

2216.397  2216.399 
2766.604 

2/23.425  2723.422 

3879.000  3879.002 


14662.885  14662.889 

*  Die  kleinc  Sfi-yercr  Flasis.     Speyer,  i8?z.          t  Astron.  A'ac/ir.,  No.  1462. 


358  THE   ADJUSTMENT   OF   OBSERVATIONS. 

"  The  remarkable  agreement  which  the  two  operations 
present  is  sufficient  authority  to  limit  the  length  of  bases, 
and  to  be  satisfied  with  those  of  2  or  3  kilometres  in  length, 
always  on  the  condition  that  they  are  joined  to  the  sides  of 
the  main  triangulation  by  means  of  a  system  of  lines  ar- 
ranged so  as  to  be  able  to  apply  to  it  the  proper  method  of 
adjustment." 

If  errors  arising  from  the  angles  of  the  triangles  are 
neglected,  it  is  easy  to  show  that  if  a  short  base  M  is 
measured  n  times,  and  the  arithmetic  mean  of  the  n  meas- 
urements taken  as  base  from  which  a  line  equal  to  nM  is 
derived  by  triangulation,  the  precision  of  this  line  is  the 
same  as  if  it  had  been  measured  once  directly.* 

For  if  [JLM=  the  m.  s.  e.  of  a  length  M,  then  f*MV7i  is  the 
m.  s.  e.  of  a  length  nM.  Also,  the  m.  s.  e.  of  the  arithmetic 

mean   of  n   measurements  of  the  line  M  is  —7^,  and  when 

Vn 

from  this  line  the  line  nM  is  derived  trigonometrically  its 
m.  s.  e.  is  —.--  n  =  HM  Vnt  agreeing  with  the  preceding. 

V       Jl 

Since,  however,  the  principal  errors  arise  from  the  tri- 
angulation, the  advantage  is  with  the  longer  base. 

We  may  estimate  the  relative  amount  of  influence  of 
errors  in  the  base  and  of  errors  in  the  angles  in  a  chain  of 
triangles,  on  the  value  of  any  side  computed  from  the  base, 
as  follows : 

Assuming  the  triangles  to  be  approximately  equilateral, 
we  have  (Ex.  9,  Art.  in) 

Ha*  —  /V  -J-  f  //  sin2 \"  Vn 

where  aH  is  the  computed  side,  b  the  base,  n  the  number  of 
triangles  intervening,  and  //  the  m.  s.  e.  of  a  measured 
angle. 

Suppose,  for  the  sake  of  fixing  our  ideas,  that  b  —  10000™, 
then 

ua^  =  fj-b  +-1600  nfjf  millimetres. 

*  Gradmessung  in  Ostpreussen,  p.  36. 


APPLICATION   TO    BASE-LINE    MEASUREMENTS.  359 

Now,  with  a  primary  base  apparatus  a  precision  of  a 
m.  s.  e.  of  2  mm.  in  icoo  m.  can  be  easily  reached.*  Hence 
from  the  above  formula  we  see  that  even  in  a  short  chain 
of  triangles  the  error  of  a  side  arising  from  the  error  of  the 
base  is  small  in  comparison  with  that  arising  from  the  errors 
in  the  angles  measured. 

Also,  since  the  m.  s.  e.  of  a  side  arising  from  errors  in 
the  angles  measured  increases  as  the  square  root  of  the 
number  of  triangles  from  the  base  (Ex.  9,  Art.  in),  we  con- 
clude that  it  is  better  to  measure  bases  frequently  with 
moderate  precision  rather  than  to  measure  a  few  at  long 
intervals  in  the  net,  but  with  great  precision. 

But  little  is  gained  in  precision  by  repeating  the  meas- 
urement of  the  base  many  times.  We  have  seen  in  Art. 
164  that  the  main  sources  of  error  to  be  leared  arise  from 
the  comparisons  with  the  standards  and  are  independent  of 
the  measurement  of  the  line.  Accordingly,  though  bases 
have  in  the  present  century  been  measured  from  6  to  8 
times,  it  is  the  general  custom  now  to  do  so  only  twice. 
In  this  way  any  gross  error  is  checked  and  a  sufficiently 
close  precision  determination  of  the  measurement  can  be 
found.  As  a  compromise  between  the  error  arising  from 
the  connection  with  the  triangulation  and  the  systematic 
error  introduced  by  the  base  apparatus  itself,  the  general 
practice  is  to  measure  bases  of  from  4  to  8  kilometres  f  in 
length,  or  about  one-sixth  of  the  triangle  sides.  Besides, 
long  bases  are  not  always  to  be  had  in  positions  .just  where 
wanted  as  triangle  sides,  as  the  configuration  of  most  coun- 
tries will  not  allow  of  it. 

*  The  p.  e.  of  the  Madridejos  Base  (1858),  Spain,  is  —        —  part  ;  of  the  Grossenhain  Base 

5865800 

1872),  Saxony,  —       —  part  ;  of  the  Atlanta  Base  (1872-1873),  U.  S.,  part  ;  of  the  Chicago 

Base  (1877),  U.  S., part ;  of  the  Sanduskv  Base  (1878),  U.  S., part. 

1052200  1148600 

t  For  example,  Salisbury  Plain  (1849),  England,  u.i  kil.  ;  Halland  (1863),  Sweden,  7.3  kil.  ; 
Oran  (1867),  Algeria,  9.4  kil.  ;  Harlem  (1868-1869!,  Holland,  60  kil.  ;  Atlanta  (1872-1873),  U .  S., 
9.3  kil.  ;  Radautz  (1874),  Austria,  4.6  kil.  ;  Udine  (1874),  Italy,  3.2  kil.  ;  Chicago  (1877),  V.  S.,  7.5 
kil.  ;  Vich  (1877),  Spain,  2.5  kil.  ;  Gottingen  (1880),  Germany,  5.2  kil.  ;  Meppen  (1883),  Germany, 
7.0  kil. 


360  THE   ADJUSTMENT   OF   OBSERVATIONS. 

166.  Measuring  a  base  line   is  not  necessarily  a  difficult 
operation.     The  great  trouble  with  all  forms  of  apparatus 
hitherto  employed  arises   from   temperature  changes.     But 
with    the    apparatus    proposed    by   Mr.  E.   S.   Wheeler,    in 
which  the  measuring  bar  is  simply  a  bar  of  metal  packed  in 
melting  ice,  this  difficulty  would  be  altogether  overcome. 
The  length  of  the  bar  would  remain  unchanged  throughout 
the   measurement,  as  its  temperature  is  kept  constant,  being 
that  of  melting  ice.     The  same  temperature  could  at  any 
time  be  had  at  which  to  find  the  length  of  the  bar  in  terms 
of  the  official  standard  of  length. 

The  apparatus  might  be  constructed  as  follows:  The 
measuring  bar,  a  bar  of  steel  25  mm.  in  diameter  and  6m.  in 
length,  placed  in  a  circular  cast-steel  tube  I  m.  in  diameter, 
made  stiff  by  bracing,  but  as  light  as  possible.  Along  the 
top  of  this  tube  slots  of  about  75  mm.  in  width  would  be  cut 
to  allow  the  introduction  of  ice  around  the  bar.  The  hole 
for  drainage  would  be  at  the  centre  of  the  tube  on  the 
under  side.  For  supports  during  the  measurement  two 
trestles  placed  i-I  m.  from  the  ends  would  be  best.  Effects 
of  flexure  would  be  got  rid  of  by  having  the  graduation 
marks  showing  the  length  of  the  bar  placed  on  the  neutral 
axis  of  the  bar.  The  reading  microscopes,  alignment  appa- 
ratus, sector  and  level  for  determining  the  inclination  of  the 
bar  during  measurement,  such  as  those  made  by  Repsold 
for  the  U.  S.  Engineers.  The  mode  of  measurement  the 
same  as  with  the  Repsold  apparatus.*  The  amount  of 
computation  necessary  to  reduce  the  measurements  made 
in  this  way  would  be  small  in  comparison  with  that 
required  with  the  forms  of  apparatus  at  present  in 
use. 

167.  Connection  of  a  Base  with   the  Main   Tri- 
aiigulatioii. —  We  shall  now  consider  the  best  method  of 
connecting  a  base  line  with  a  side  of  the  main  triangulation 
with  the  least   possible   loss  of  precision.     This,  like  most 

*  See  Professional  Papers  Corps  of  Engineers  U.  S.  A.,  No.  24,  for  description  of  the  Rep- 
sold base  apparatus. 


APPLICATION   TO   BASE-LINE   MEASUREMENTS.  361 

geodetic  questions,  is  not  to  be  decided  by  least-squares' 
methods  alone. 

There  are  two  points  to  be  considered— as  little  loss  of 
precision  as  possible,  combined  with  economy  of  work  in  the 
measurement  and  reduction  of  the  intervening  triangula- 
tion.  To  secure  the  first  we  must  use  only  well  shaped 
triangles,  avoiding  in  particular  very  acute  angles  opposite 
the  bases.  To  secure  the  second  as  few  stations  as  possible 
should  be  occupied. 

The  solution  is  a  tentative  one.  Various  forms  of  con- 
nection may  be  tried,  and  the  m.  s.  e.  of  the  triangle  sides 
computed  from  the  bases  by  the  methods  of  Chapter  V.,  on 
the  hypothesis  of  a  regularity  of  figure  which  conforms 
more  or  less  closely  to  the  case  in  hand.  Many  of  the  re- 
sults will  be  found  in  Ex.  10-15,  Art.  in.  These  results  we 
shall  now  make  use  of. 

Taking  the  length  of  a  triangulation  side  to  be  from  15 
to  30  miles,  as  giving  the  best  results  in  angle  measurements, 
the  proper  length  of  the  base  would  be  from  3  to  5  miles— 
that  is,  about  one-sixth  of  a  triangle  side.  Now,  if  the  con- 
nection between  the  base  AB  and  a  triangle  side  6  times 
the  base  is  through  a  chain  of  equilateral  triangles  of  the 
form  of  Fig.  12  ;  or  of  the  rhomboidal  form  of  Fig.  15,  com- 
posed of  two  similar  triangles,  in  which  the  first  diagonal 
BB'  is  6  times  the  base  ;  or  of  the  rhomboidal  form  of  Fig.  16, 
in  which  the  second  diagonal  .is  6  times  the  base,  the  re- 
spective weights  of  the  derived  side  are,  roughly,  as  the 
numbers  16,  3,  7. 

Hence,  so  far  as  the  precision  is  concerned,  the  advantage 
is  with  the  first  form.  When,  however,  we  consider  that 
this  form  requires  12  stations  to  be  occupied,  while  the  third 
needs  only  6,  we  see  that  when  amount  of  labor  as  well  as 
degree  of  precision  is  taken  into  account  the  advantage  is 
with  the  latter  form.  The  second  form,  though  the  most 
economical,  is  condemned  by  the  acute  angles  that  occur  in 
it  opposite  to  the  base. 

Similar  comparisons   of  other  forms  of  connection  will 


362 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


lead  us  to  the  conclusion  that  the  rhomboidal  form  (Fig.  16), 
in  which  a  side  equal  in  length  to  about  6  times  the  base 
can  be  found  by  passing  through  two  sets  of  similar  tri- 
angles from  the  base,  is  the  normal  form,  and  that  the  forms 
used  in  practice  should  approximate  to  it  as  closely  as  the 
nature  of  the  country  will  allow.  As  examples  take  the 
connections  of  the  Coast  Survey  Atlanta  Base  (1872-1873), 
the  Lake  Survey  Chicago  Base  (1877),  and  the  Prussian 
Gottingen  Base  (1880). 

Fig.50 


Fig.49 


Fig.48 


ATLANTA 


QOTTINQEN 


The  form  of  the  triangulation  net  to  connect  the  base 
with  the  main  triangulation  being  decided  on,  it  becomes  a 
question  (but  only  a  secondary  one)  to  decide  with  what 
care  the  several  angles  in  this  net  should  be  measured  in 
order  to  attain  the  greatest  precision  with  a  given  expendi- 
ture of  labor.  This  will  be  found  fully  discussed  in  ZeitscJir. 
fiir  Vermess.,  1882,  pp.  122  seq. 

1 68.  Adjustment  of  a  Triaiigulatioii  when  more 
than  one  Base  is  considered. — In  all  discussions  so  far 
we  have  considered  only  one  measured  base.  But  in  a  net 
of  triangles,  bases  must  be  measured  at  intervals,  for  reasons 
already  assigned.  In  computing  an  intermediate  side  from 
different  bases  discrepancies  will  be  found.  How  shall 
these  discrepancies  be  treated? 

The  triangulation   adjustment  could  have  been  made  in 


APPLICATION   TO   BASE-LINE    MEASUREMENTS.  363 

such  a  way  that  no  discrepancy  would  show  itself  in  pass- 
ing from  one  measured  base  to  another.  It  was  only  neces- 
sary to  introduce  equations  connecting  the  bases  in  the 
form  of  an  ordinary  side  equation,  thus  (Fig.  12), 

a       sin  A    sin  A 


b       sin  B^  sin  B^ 

where  a,  b  are  two  bases,  and  Alt  B^  .  .  .  are  the  angles  of 
continuation. 

There  is  no  objection  to  doing  this  so  long  as  the  bases 
have  been  measured  with  the  same  apparatus  and  the  inter- 
vening triangulation  is  first-rate.  But  as  the  discrepancy 
between  the  measured  value  of  the  base  and  its  value  as 
computed  from  another  base  through  the  triangulation 
affords  a  good  test  of  the  quality  of  the  work,  it  is  better 
not  to  introduce  an  equation  connecting  the  bases  into  the 
first  adjustment. 

169.  Suppose  that  the  triangulation  has  been  adjusted  in 
sections,  each  with  reference  to  a  single  measured  base,  as 
explained  in  Chapter  VI.,  and  we  wish  to  adjust  for  the 
discrepancy  arising  from  computing  one  base  from  another 
through  a  chain  of  the  best-shaped  triangles  in  the  system. 
It  will  be  sufficient  to  confine  our  attention  to  this  chain  of 
triangles,  all  tie  lines  being  rejected. 

(i)  Rigorous  Solution.  —  The  condition  equation  to  be 
satisfied,  arising  from  the  connection  of  the  bases,  may  be 
written 

<*  +  (*)       sin  \A,  +  (Aft  sin  \At+(At)\ 
b  +  (b]       sin  {/?,+(#,)}  sin  {*, 

where  a,  b,  At,  Z?,,  .  .  .  are  measured  values,  and  (a),  (/;), 
(/?,),  (-5,),  .  .  .  are  their  most  probable  corrections. 

Taking  logs,  and  reducing  to  the  linear  form  (Ex.  4, 
Art.  64), 

-  d.(a)  +  W)  +  [3A(A)  -  J^)]  =  / 

47 


364  THE  ADJUSTMENT   OF   OBSERVATIONS. 

where  /  is  the  excess  of  the  observed  over  the  computed 
value  of  log  a,  and  da,  ob,  dA,  dB  are  the  log.  differences  as 
usual. 

Also,  since  the  angles  of  each  triangle  in  the  chain  must 
satisfy  the  condition  of  closure,  we  have  the  conditions 


with 

°  =  a  mn- 


\  ] 


where  /^a,  /Jtb  are  the  m.  s.  e.  of  the  bases,  and  //,,  //2,  .  .  .  the 
m.  s.  e.  of  the  three  angles  of  each  of  the  triangles  in  the 
chain. 

The  solution  may  be  carried  out  by  the  method  of  cor- 
relates. 

If  the  bases  have  been  measured  with  different  apparatus 
the  question  of  the  comparison  of  the  measuring  bars  with 
the  standards  of  length  is  the  important  one.  After  that 
has  been  satisfactorily  settled  the  connection  of  the  bases 
may  be  treated  as  above. 

Ex.  This  mode  of  reduction  maybe  illustrated  by  the  simple  case  of  a 
triangle  having  two  sides  and  the  three  angles  measured. 

In  the  triangle  W.  Base,  E.  Angle,  W.  Angle  (ABC),  Sandusky  Base, 
there  were  measured 

ft.  ft. 

BC  =  6742.420  ±  o.oio 

AC  =  6602.386  ±  o.oio 

From  the  adjustment  of  the  triangulation  with  reference  to  one  measured 
base,  and  which  was  carried  out  by  the  methods  already  explained  in  Chapter 
VI.,  the  three  adjusted  angles  of  the  triangle  were  found  to  be 


i°  07'  51".  35  ±  o".2o 
ABC=  i°  06'  26".  74  ±  o".20 
ACB  =  177°  45'  41".  91  ±  o".2o 

Required    the   most   probable  values  of  the  sides  and  angles  of  the  triangle 
from  a  second  adjustment  into  which  the  two  measured  sides  enter. 


APPLICATION   TO   BASE-LINE    MEASUREMENTS.  365 

If  (a),  (/>),  (A),  (B),  (C)  arc  the  corrections  to  the  above  quantities  in  order, 
the  condition  equations  are 

(A)  +  (B)  +  (C)  =  o 

6742.420  +  (a)  _  sin  \  i°  07'  51".  35  +  (A)\ 
6602.386  +  (/>)   ~  sin  |  r°  06'  26".  74  4-  (B)\ 

or,  reducing  the  latter  to  the  linear  form, 

—  o.644(<z)  +  0.658(6)  —  i.oS()(A)  +  i.obi(B)  —  —  0.044 
with 

(a)*          W_    ,     W    .     (JBT-        (C)2 
(.01)-  +  (.01)*  +  (.20)-  +  (.20)*  +  (^  - 

that  is, 

4oc<rt)2  +  40o(/')2  +  (A)-  +  (B)-  +  (C)-  =  a  min. 

The  solution  of  these  equations  gives 

ft. 

(a)  =  +  0.00003         (^)  =  —  o".O2 

(t>)  =  —  0.00003         (^)  =  +  o".O2 

(C)  =       o".oo 

This  example  is  noteworthy  as  showing  the  combination  of  heterogeneous 
measures  in  the  same  minimum  equation. 

(2)  Approximate  Solutions  —  The  two  of  most  importance 
are  (a)  when  the  angles  alone  are  changed,  (b)  when  the 
bases  alone  are  changed.  Either  of  these  is  practically 
more  important  than  the  rigid  solution,  as  a  base  line  in 
good  work  receives  a  very  small  correction  from  the  ad- 
justment. 

(a)  When  the  angles  alone  are  changed.  The  formulas 
to  be  used  in  this  case  follow  from  those  of  the  rigorous 
method  just  given  by  putting  the  base  corrections  equal  to 
zero.  Thus,  if  all  of  the  adjusted  angles  are  of  the  same 
weight,  then,  since  (a]  =  o,  (//)  —  o,  the  baseline  equation 
becomes 


with 

The  angle  equations  are  as  before. 


366  THE   ADJUSTMENT   OF    OBSERVATIONS. 

Hence  if  k  is  the  correlate  of  the  base-line  equation,  we 
have,  by  eliminating  the  angle  equation  correlates, 


(Q=-    (V-     dB'}k 

(2) 
(A,}=        (20Y  +    dB"}k 


=  -  OY-  V)* 
•        «        .        .        . 

whence,  by  substituting  in  the  base-line  equation, 


Hence  the  corrections  to  the  angles  are  known. 

A  still  further  approximation  may  be  made.  It  is  evident 
that  the  corrections  to  the  angles  C  are  small  compared 
with  those  to  A  and  B,  and  that  they  vanish  when  A=B. 
Hence,  as  we  have  assumed  the  triangles  to  be  well  shaped, 
we  may  take 

=  (Q=.  .  .  =o  (3) 


The  angle  equations  then  become 
whence 


.^_        '^B_k  .  (5) 

2 

where 


APPLICATION   TO    BASE-LINE   MEASUREMENTS.  367 

Ex.  i.  To  find  the  changes  in  the  angles  resulting  from  the  equation 
connecting  the  lengths  of  the  lines  1-2  and  14-15  in  the  triangulation  of  Long 
Island  Sound  (Fig.  46). 

The  excess  of  the  log.  of  the  observed  v.ilue  of  14-15  over  the  value- 
computed  from  1-2  through  the  triangulation  is  4.00  in  units  of  the  sixth 
decimal  place. 

If  the  lines  themselves  receive  no  correction  the  condition  equation, 
expressed  in  the  linear  form,  is  (see  table,  p.  346) 


0.27(^1)  —  o.99(j9i)  +  i.6i(/la)  —  o.6o(B-i)  +  .  .  .  —4.00 
Now, 

[(8  A  +  sBy>]  =  51.48 

and  therefore 

0.27  +  o.qq 
(Al)  =  -(Bl)  =  —  ^X  4-00=0".  10 


the  corrections  required. 

Ex.  2.  To  find  the  precision  of  a  side  in  a  chain  of  triangles  joining  two 
bases. 

Having  found  the  adjusted  angle?,  we  may  find  the  weight  of  the  log.  of 
any  side  of  continuation  as  the  mth  in  a  chain  of  «  triangles  joining  two  bases, 
all  of  the  angles  being  of  the  same  weight. 

For  simplicity  in  writing  take  in  =2,  n  —  3. 

The  condition  equations  are 


3)  -  8B"(B,)  +  8A'"(A3)  -  8B"\B,) 
and  the  function  /',  whose  weight  is  to  be  found, 

F=8A'(Ai)  -  8B'(B^  +  8A"(A9)  -  8B"(B*) 
The  solution  follows  at  once  from  Eq.  15,  Art.  in.     The  result  is 

a  8A'"*  +  8A"   SB"  +  8B"'- 

UF=  f  [8A>  +  8A  dB  +  8B*]  ~         , 

[<>A*  +  t>A  8B  +  6B-]t 

the  summation  of  the  8's  extending  between  the  limits  indicated. 


368  THE   ADJUSTMENT   OF   OBSERVATIONS. 

As  an  example  of  a  different  method  of  treating  this 
special  case  (a),  the  Coast  Survey  connection  of  three 
primary  bases,  the  Fire  Island,  Massachusetts,  and  Epping, 
may  be  cited.* 

The  triangulation  net  connecting  the  bases  is  conceived 
to  consist  of  three  branches,  one  for  each  base  and  pro- 
ceeding therefrom  to  a  line  in  common.  Adjusting  each 
branch,  three  independent  values  of  the  length  and  p.  e.  of 
the  line  of  junction  are  obtained.  The  weighted  mean  of 
these  values  is  taken  to  be  the  most  probable  value  of  the 
line  of  junction.  This  value,  as  well  as  the  length  of  each 
base,  is  considered  to  be  exact.  The  adjustment  of  each 
branch  of  the  triangulation  is  then  repeated  with  an  addi- 
tional equation  fixing  the  ratio  of  the  length  of  the  base  to 
the  line  of  junction.  In  making  this  adjustment  the  form 
of  solution  given  in  Art.  90  will  be  found  very  convenient 
in  handling  the  extra  equation. 

In  the  final  computation  of  the  lengths  of  the  triangle 
sides  "from  any  one  of  the  measured  base  lines,  we  shall, 
as  we  recede  from  it,  obtain  a  proportional  amount  of  the 
influence  of  the  measure  of  the  other  two  lines  as  we  ap- 
proach them,  and  finally,  reaching  one  or  the  other,  the  effect 
of  that  base  from  which  we  set  out  is  lost.  Each  distance 
will  have  assigned  to  it  its  most  probable  value ;  we  shall 
have  no  discrepancy  whatever  in  the  geometrical  figure  of 
the  triangulation,  and  the  resulting  sides  and  angles  will 
have  nearly  the  same  probability  as  those  derived  from  a 
theoretically  perfect  solution." 

(b)  By  changing  the  bases  alone.  The  argument  for 
changing  the  bases  only  is  that  as  the  computed  discrep- 
ancy is  always  small,  and  as  the  triangulation  has  been 
already  adjusted,  it  is  a  great  saving  of  labor  to  change 
all  the  triangles  proportionally,  and  thus  get  a  consistent 
result  by  a  method  which,  if  not  rigorous,  is  good  enough. 
This  is  the  more  evident  if  we  consider  with  what  accuracy 
geodetic  work  can  now  be  done. 

*  Report,  1865,  app.  No.  21. 


APPLICATION   TO    BASE-LINE    MEASUREMENTS.  369 

Let  £,,  b.»  ...  be  the  measured  lengths  of  the  bases,  and 
/,,  Pv  .  .  .  their  weights. 

If  s  is  the  most  probable  length  of  a  common  side  com- 
puted from  the  bases,  and  /,,  /2,  .  .  .  are  the  ratios  of  the 
lengths  of  the  bases  to  this  side,  then 

s  =  y      weight  p}>?,  and  weight,  of  log  y1  is  pjb* 
/ ,  X , 

*  =  r  M'  log  £  is  /.A* 

' v  <i  A  „ 


The  weighted  mean  value  of  log  s  is 


With  this  value  of  log  s,  leaving  the  angles  of  the  triangles 
as  they  are,  the  sides  may  be  recomputed,  when,  of  course, 
the  bases  will  be  changed. 

We  have  seen  in  Art.  164  that  in  base-line  measurements 
the  error  arising  from  errors  of  comparison  of  the  measuring 
bars  is  the  main  one,  and  that  the  error  arising  from  the 
measurement  of  the  line  itself  is  relatively  small.  Hence 
we  may  take  the  m.  s.  e.  of  a  base  as  proportional  to  the 
length,  and  the  weight,  therefore,  inversely  as  the  square 
of  the  length.  The  weight  of  each  computed  value  of  log  s 
will  then  be  the  same,  and  we  have  now 


As  the  weights  taken  are  tolerably  close,  this  simple  formula 
will  give  a  result  near  enough  for  most  work. 

It  is  evident  that  if  we  compute  each  base  from  all  of 
the  others,  and  take  the  mean  of  the  logarithmic  values 
found,  consistent  values  of  the  sides  throughout  the  tri- 
angulation  will  be  found  by  starting  from  any  base. 


370  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Thus  in  the  Coast  Survey  example  already  referred  to, 


Epping  Base. 

Mass.  Base. 

Fire  Island  Base. 

Measured, 

3.9403143.4 

4.2387077.4 

4-1479535-3 

Computed  from  Epping, 

4.2387115.5 

4.1479556.3 

"             "      Mass., 

3.9403075.8 

4.1479518.4 

"      Fire  Id., 

3.9403122.4 

4.2387094.6 

Means, 

3.9403113.9 

4.2387095.8 

4.1479536.7 

There  will  now  be  no  contradiction  in  the   values  of  any 
triangle  side  deduced  from  the  different  bases. 


CHAPTER  VIII. 

APPLICATION   TO   LEVELLING. 

170.  The  object  of  levelling  is  to  find  the  difference  of 
height  of  two  or  more  points  on  the  surface  of  the  earth. 
As  only  differences  of  height,  and  not  absolute  values,  are 
required,  some  one  height  may  be  selected  as  the  standard 
of  reference,  and  to  it  any  value  at  will  may  be  assigned. 
The  common  custom   in  geodetic  work  is  to  take  what  is 
called  the  mean  level  of  the  ocean  as  the  standard  height. 
To  be  able  to  refer  to  it  at  any  time  a  fixed  mark  is  estab- 
lished, say  at  New  York,  and   readings  are  taken  to  the  sur- 
face of  the  ocean  at  high  and  low  water  for  a  lunation  at 
least.     The  height  of  the  fixed   mark  above  mean   tide  is 
computed  from  these  readings.     Then  all  that  is  to  be  done 
in  order  to  find  the  elevation  of  any  point  above  mean  tide 
is  to  run  a  line  of  levels  between  this  point  and  the  fixed 
mark  just  described. 

There  are  two  methods  of  levelling  in  common  use  in 
geodetic  work  :  spirit  levelling  of  precision-and  trigonomet- 
rical levelling.  The  difference  between  the  two  consists 
mainly  in  the  greater  lengths  of  lines  sighted  over  in  the 
latter  case. 

Spirit  Levelling  of  Precision. 

171.  For  descriptions  of  instruments  employed  see  AY- 
vellement   de   Precision   de   la  Snisse,    par  Hirsch    et    Plnnta- 
mour,  Geneve  et  Bale,  1867,  seq.  ;  Report  Chief  of  Engineers 
U.  S.  A.,    1880,  App.   GO;  Report  U.  S.  Coast   Survey,    18/9 
App.   15,   1880  App.   II  ;   Professional  Papers  Corps  of  Engi- 
neers U.  S.  A.,  No.  24,  Chap,  xxii 

4s 


372  THE  ADJUSTMENT   OF   OBSERVATIONS. 

The  method  of  observing  used  on  the  Coast  Survey  is 
as  follows:  Two  levelling  rods  are  used.  "Two  lines  are 
run  simultaneously,  with  the  rods  usually  at  different  dis- 
tances from  the  instrument ;  and  to  prevent  the  gradual 
accumulation  of  error  supposed  to  be  due  to  running  con- 
stantly in  one  direction,  alternate  sections  are  run  in  opposite 
directions.  Each  station  of  the  instrument,  therefore,  con- 
tains two  backsights  and  two  foresights.  In  stations  i,  3, 
5,  etc.,  rod  A,  backsight,  is  invariably  read  first ;  then  rod  B, 
backsight;  then  rod  A,  foresight;  and  finally  rod  B,  fore- 
sight. In  stations  2,  4,  6,  rod  B  takes  precedence  of  A  in 
both  back  and  fore  sights." 

An  important  source  of  error  in  spirit  levelling,  and  one 
very  commonly  overlooked,  is  the  change  in  the  length  of 
the  levelling  rod  from  variations  of  temperature.  From 
experiments  made  by  the  Prussian  Land  Survey,  in  which 
the  rods  were  compared  daily  with  a  steel  standard,  the 
following  fluctuations  in  length  were  found  for  four  rods 

*~*  o 

made  of  seasoned  fir :  * 

Rod  13,  from  May  19  to  Aug.  18,  0.51  mm.  per  metre. 
14,     "          "     20  "      "      15,  0.46     " 
9,     "          "     24  "  Sept.    6,  0.37     " 
10,     "          "     24  "      "       6,  0.43     " 

It  is  quite  ppssible  that  errors  from  this  source  may 
largely  exceed  the  errors  arising  from  the  levelling  itself. 
Each  field  party  should  therefore  be  provided  with  the 
means  of  making  a  daily  comparison  of  the  rods  used  with 
a  standard  of  length.  A  steel  metre  and  a  micrometer 
microscope  mounted  on  a  stand  would  be  all  that  would  be 
necessary. 

It  is  a  curious  fact  in  spirit  levelling,  but  one  abundantly 
verified,  that  a  much  greater  discrepancy  is  to  be  expected 
in  a  duplicate  line  of  levels  if  the  lines  have  been  levelled 
over  in  opposite  directions  than  if  both  have  been  run  in 
the  same  direction.  This  was  noticed  long  since  in  a  line 

*  Nivellements  dcr  Lamtesati/nafime,  vol.  v.     Berlin,  1883. 


APPLICATION   TO    LEVELLING. 


373 


of  levels  run  under  the  direction  of  a  committee  of  the 
British  Association  from  the  Bristol  Channel  to  the  English 
Channel  (Portishead  to  Axraouth),  a  distance  of  74  miles. 
The  increase  in  the  discrepancy  of  the  forward  and  back- 
ward levels  was  continuous.  Thus  at  12  miles  from  Portis- 
head it  was  0.35  ft.,  at  23  miles  0.53  ft.,  at  37  miles  0.70  ft., 
at  49  miles  0.82  ft.,  at  59  miles  0.92  ft.,  and  at  74  miles 
1.03  ft.* 

In  the  winter  of  1878-1879  a  line  of  levels  was  run,  under 
the  direction  of  the  U.  S.  Engineers,  from  Austin,  Miss.,  to 
Friar's  Point,  Miss.,  a  distance  of  about  44  kilometres. 
"  All  lines  were  levelled  in  duplicate  and  in  opposite  direc- 
tions. When  using  two  rods  both  were  used  on  the  same 
line,  rod  No.  2  being  used  for  the  first,  third,  .  .  .  back- 
sight and  for  the  second,  fourth,  .  .  .  foresight,  and  rod 
No.  3  for  the  second,  fourth,  .  .  .  backsight  and  for  the 
first,  third,  .  .  .  foresight,  always  using  rod  No.  2  on  the 
closing  bench  mark."  The  results  for  the  principal  bench 
marks  are  given  in  the  following  table,  where  it  will  be 
seen  that  the  discrepancy  in  the  forward  and  backward 
levels  increases  from  one  end  of  the  line  to  the  other:  f 


Distance. 

Bench  Mark. 

Discrepancy. 

Austin  I. 

m 

mm 

16505 

Trotter's  Landing. 

+  16.8 

4464 

Glendale. 

-f  16.8 

15950 

Delta. 

+  23.6 

6602 

Friar's  Point  I. 

+  36.5 

Experience  on  the  survey  of  the  great  lakes  and  of  the 
Mississippi  River,  and  also  on  the  survey  of  India,  has 
shown  that  the  personal  peculiarities  of  the  observer  enter 


Report  lifit ish  Association, 


t  Re  fort  Chit, 


Engineers  I',  S.  A.,  1879,  P-  T944- 


374  THE  ADJUSTMENT   OF   OBSERVATIONS. 

largely  into  levelling  work.  It  would  seem  that  a  much 
more  reliable  result  will  be  obtained  if  the  line  between  two 
points  whose  difference  of  height  is  required  has  been  run 
over  not  only  in  opposite  directions,  but  in  opposite  direc- 
tions by  the  same  observer.  The  personal  bias  is  probably 
due  most  largely  to  the  fact  that,  the  ends  of  the  bubble 
not  being  sharply  defined,  different  observers  estimate  the 
positions  of  these  end  points  differently.  To  eliminate  its 
effect  completely  the  line  should  be  levelled  in  opposite 
directions  by  each  of  a  large  number  of  observers. 

Again,  as  both  rod  and  bubble  should  be  read  simul- 
taneously at  each  foresight  and  at  each  backsight,  and  as 
this  is  physically  impossible  for  one  observer,  the  effect  of 
the  unequal  heating  of  the  instrument  by  the  sun,  even 
when  shaded  by  an  umbrella,  has  a  tendency  to  cause  a 
change  in  the  position  of  the  bubble  in  the  interval  between 
the  readings  of  the  rod  and  bubble.  This  error  is  cumula- 
tive, and  its  reduction  to  a  minimum  depends  upon  the  skill 
of  the  observer. 

The  rule  adopted  by  the  European  Gradmessung  for 
allowable  discrepancy  in  a  duplicate  line  of  levels  is  that 
the  p.  e.  of  the  difference  in  height  of  two  points  one  kilo- 
metre apart  should  in  general  not  exceed  3  mm.,  and  should 
in  no  case  exceed  5  mm.  On  the  U.  S.  Coast  Survey,  for 
short  distances  a  discrepancy  between  two  levellings  of 
a  distance  of  D  kilometres  of  an  amount  not  exceeding 
5  \^2D  mm.  is  allowed. 

172.  Precision  of  a  Line  of  Levels. — The  precision 
of  a  line  of  levels  will  be  given  by  the  m.  s.  e.  of  a  single 
levelling  of  a  unit  of  distance,  which  we  shall  take  to  be  one 
kilometre.  The  problem  is  quite  analogous  to  that  already 
discussed  in  Art.  164,  the  unit  of  distance  there  being  the 
length  of  a  measuring  bar. 

If  we  suppose  that  in  running  a  single  line  of  levels  be- 
tween two  bench  marks,  A  and  B,  the  ground  is  equally 
favorable  throughout,  and  that  none  but  accidental  errors 
ma}'  be  expected  to  enter,  then  if  //  is  the  m.  s.  e.  of  the 


APPLICATION    TO    LKVELUNO.  37$ 

unit  of  distance  (one  kilometre),  the  m.  s.  e.  of  a  distance  of 
D  kilometres  would  be 


in   other  words,  the  weight  of  a  levelling  of  a  distance  of  D 
kilometres  would  be  invt  rsely  proportional  to  tliat  distance. 

Strictly  speaking,  we  should  find  the  weight  from  the 
m.  s.  e.  //„  //„  .  .  .  arising  from  all  sources  of  error  that 
enter  into  the  work,  such  as  from  the  comparisons  of  the 
levelling  rod  with  the  standard,  from  the  nature  ot  the 
country  levelled  over,  from  effects  of  change  of  length  of 
sight,  etc.  Then,  considering  the  errors  arising  from  these 
sources  to  be  independent,  we  should  find  finally  the  m.  s.  e. 
of  each  line  levelled  over  to  be  vf/?j,  and  the  weight  of  the 

line  would  therefore  be  as  T 

O'J 

The  uncertainty  attendant  on  estimating  these  sources 
of  error  is  so  great,  and  the  influence  of  the  distance  so 
largely  exceeds  the  influence  of  the  others,  that  it  is  suffi- 
cient to  adopt  the  rule  first  given  of  weighting  as  the  in- 
verse distance.  It  follows  from  this  rule  that  better  work 
is  to  be  looked  for  if  short  sights  are  taken.  Even  with  a 
first-rate  telescope  sights  should  not  be  taken  to  exceed  100 
metres.*  As  the  rod  is  more  easily  read  by  the  observer  at 
short  distances,  the  loss  of  time  from  the  more  frequent 
settings  of  the  instrument  is  not  so  great  as  would  at  first 
appear. 

Suppose  now  that  in  finding  the  difference  of  height  of 
two  bench  marks,  A  and  B,  »,  lines  of  levels  have  been  run, 
and  that  the  results  have  been  compared  at  intermediate 
bench  marks  in  succession  Dlt  Dv  .  .  .  Dn  kilometres  apart; 

*  On  the  precise  levelling  of  the  Coast  Survey,  "  where  the  slope  of  the  ground  is  steep  the 
distances  may  be  taken  as  great  as  possible,  and  on  comparatively  level  ground  they  may  range 
from  50  to  150  metres,  according  to  the  condition  of  the  weather  and  atmosphere." 

The  U.  S.  Engineers  follow  the  rule,  "  The  lengths  of  sight  will  depend  on  the  condition  of 
the  atmosphere,  but  the  rod  should  always  be  near  enough  to  be  seen  distinctly.  It  will  be  seldom 
that  lengths  of  sight  greater  than  150  metres  can  be  taken." 

In  the  Prussian  Land  Survey  "the  length  of  sight  since  1879  has  not  beon  taken  over  50". 
Only  in  special  cases  —  for  example,  in  crossing  streams  —  is  it  permitted  to  exceed  50",  whereas  a 
shorter  sight  is  frequently  taken." 


3/6  THE   ADJUSTMENT   OF   OBSERVATIONS. 

then  if  z//,  v",  .  .  .  ;  v,',  vt",  ...;...  denote  the  residual 
errors  of  the  differences  of  height  in  the  n  sections,  and 
//,  p",  .  .  .  ;  //»  p",  ...;...  denote  the  weights  of  the 
observed  differences,  we  have,  as  in  Art.  164  — 

(i)  On  the  hypothesis  that  the  precision  for  each  unit  of 
distance  (one  kilometre)  is  the  same  throughout  the  different 
sections, 


n  (n,  -  i) 


«= 


«(*,- 

since  the  weights  are  inversely  as  the  distances. 

(2)  On  the  hypothesis  of  the  independence  of  the  sec- 
tions between  the  intermediate  bench  marks,  the  average 
value  of  fJL  is  given  by 


When  the  number  of  levellings  is  two,  and  the  ob- 
served differences  of  height  at  the  several  bench  marks 
Z>,,  Z>2,  .  .  .  Dn  kilometres  apart  give  discrepancies  of 
dlt  d^  .  .  .  dn  respectively,  then  the  above  formulas  reduce 
to 

and 


The  m.  s.  e.  of  a  single  levelling  of  the  whole  line  would  be 
respectively 


and 


and  for  the  m.  s.  e.  of  the  mean  of  the  two  measurements 
these  values  would  each  be  divided  by   V2. 


APPLICATION   TO    LEVELLING. 


377 


Ex.  i.  In  the  precise  Levelling  (1880)  of  the  U.  S.  Coast  Survey  on  the 
Mississippi  River  two  lines  were  run  simultaneously  between  eveiy  two 
bench  marks.  The  following  are  the  results  from  B.M.  LIV.  to  B.M.  LV. 


B.M. 

Distance. 

Rod  A. 

Rodfl. 

Difference. 

kil. 
3-447 

m. 
—  O.S7O2 

>«. 
.-  0.8713 

mm. 
I.I 

3.806 

+  0.5805 

+  0.573S 

6.7 

1.204 

-0.2154 

—  0.2204 

5-0 

3.038 

+  0.2667 

+  0.2664 

0.6 

It  will  be  found  that  //  =  2mm  nearly. 

Ex.  2.  The  distance  AB  has  been  levelled  n  different  limes.  Calling  di 
the  difference  between  the  first  and  second  measurements,  il3  the  difference 
between  the  first  and  third,  and  so  on,  show  that  the  m.  s.  e.  of  an  observation 
of  weight  unity  is 


n(n  —  i) 

Ex.  3.  The  difference   of  level   of  two   points,  A  and  B,  is  found  by  two 
routes  whose  lengths  are  Dt,  D-2  kil.  respectively.     If  the  discrepancy  in  the 

results  is  d,  the  m.  s.  e.  of  one  kilometre  is  -  — , 


173.  Adjustment  of  a  Net  of  Levels. — If  a  line  of 
levels  is  to  be  run  between  two  points  a  sufficient  check  ot 
the  accuracy  of  the  work  will  in  general  be  afforded  by 
running  the  line  over  at  least  twice.  Comparisons  may  be 
made  at  intermediate  points  not  too  far  apart,  and  if  the 
discrepancies  found  are  within  the  limits  already  mentioned 
the  mean  of  the  results  may  be  taken  as  giving  the  eleva- 
tions sought.  This  method  was  used  by  the  U.  S.  Engi- 
neers in  determining  the  heights  of  the  great  lakes  between 
Canada  and  the  United  States  above  mean  tide. 

But  when  a  complete  topographical  survey  of  a  country 
is  made,  and  a  network  of  levels  is  necessary,  an  additional 
control  of  the  accuracy  of  the  work  is  afforded  by  the  polyg- 
onal closing  of  the  level  lines  forming  the  net;  that  is, 
from  the  condition  that  on  passing  round  the  polygon  and 


378  THE   ADJUSTMENT   OF   OBSERVATIONS. 

arriving  at  the  starting-point  we  should  have  the  same 
height  as  at  first.  This  assumes  that  the  points  of  the  net 
are  in  the  same  level  surface  and  that  error  of  closure  de- 
pends upon  errors  of  observation  only.  In  the  usual  case, 
where  the  net  is  small  and  the  country  comparatively  level, 
such  an  assumption  is  quite  allowable.  On  the  other  hand, 
if  the  net  is  very  large  or  the  country  mountainous,  system- 
atic sources  of  error,  arising  principally  from  the  spheroidal 
form  of  the  earth  and  the  deviation  of  the  plumb-line,  would 
be  introduced.  The  corrections  resulting  from  these  causes 
would  have  to  be  computed  and  applied.  A  full  investiga- 
tion of  this  point  will  be  found  \v\Astron.  Nachr.,  Vols.  80-84. 
The  adjustment  of  a  net  of  levels  may  be  carried  out  in 
a  similar  way  to  the  adjustment  of  a  triangulation.  Thus 
suppose  that  the  lines  of  levels  form  a  closed  figure.  The 
conditions  to  be  satisfied  among  the  observed  differences 
of  height  may  be  divided  into  two  classes: 

(a)  Those  arising  from  non-agreement  of  repeated  meas- 
urements of  differences  of  height  between  successive  bench 
marks.     The  equations  expressing  these  conditions  corre- 
spond to  the  local  equations  in  a  triangulation. 

(b)  Those  arising  from  the  consideration  that  on  starting 
from  any  bench  mark  and  returning  to  it  through  a  series 
of  bench   marks,  thus  forming  a  closed  figure,  we  should 
find   the  original   height.     The  resulting  equations   corre- 
spond to  the  angle  and  side  equations  in  a  triangulation. 

If  the  circuit,  instead  of  being  a  simple  one,  has  tie  lines, 
the  number  of  closure  conditions  is  easily  estimated.  For 
if  s  be  the  number  of  bench  marks,  and  /  the  number  of 
lines  levelled  over,  the  number  of  lines  necessary  to  fix  the 
bench  marks  is  s  —  i,  and  therefore  the  number  of  super- 
fluous lines — that  is,  the  number  of  closure  conditions  to  be 
satisfied  among  the  differences  of  height — is 

/— j-f  i. 

1 74.  Approximate  Methods  of  Adjustment. — The  differences 
of  height  between  successive  bench  marks  may  be  found 


APPLICATION   TO   LEVELLING.  379 

by  taking  the  means  of  the  observed  values,  as  explained  in 
Art.  172.  Taking  these  means  as  independently  observed 
quantities,  they  may  •  be  adjusted  for  non-satisfaction  of 
closure  conditions  in  various  ways  : 

(a)  The  differences  of  level  between  the  successive  bench 
marks  will  give  rise  to  /  observation    equations.     Taking 
the  starting  point  as  origin,  .$• —  i  differences  are  required 
to  fix  the  s  stations.     If  these  s—  i    differences    are  con- 
sidered independent  unknowns  the  remaining  /  —  s-{-  i  un- 
knowns may  be  expressed  in  terms  of  them,  and  the  solution 
completed  by  the  method  of  Art.  109. 

(b)  Since  each  independent  polygon  in  the  net  contains 
one  superfluous   measurement,  it  gives  a  condition  equa- 
tion.    With  s  stations  connected  by  /  lines  there  must  be 
/—  s-\-  i  condition  equations,  as  the  number  of  superfluous 
measurements  is  /  —  s  -\-  i.     The  solution  may  be  completed 
by  the  method  of  correlates,  Art.  no. 

As  to  the  relative  advantages  of  the  two  forms  of  solu- 
tion, if  the  adjusted  values  only  are  to  be  found,  without 
their  weights,  the  test  will  be  furnished  by  the  number  of 
normal  equations  in  each  case,  as  the  principal  part  of  the 
labor  consists  in  solving  the  normal  equations.  The  num- 
ber by  the  first  form  is  s  —  i,  and  by  the  second  /—  s  -\-  i  ; 
and  therefore,  in  an  extensive  net  with  few  tie  lines,  the 
method  of  condition  equations  is  to  be  preferred,  and  vice 
versa. 

(c)  Adjust  each  simple  circuit  in  order  by  the  principle 
of  Art.  115,  and  repeat  the  process  until  the  required  accu- 
racy is  reached. 

(d)  The  method  employed  on  the  British  Ordnance  Sur- 
vey in  reducing  the  principal  triangulation.     (See  Art.  152.) 
This  method  is  easily  applied  and  gives   results  practically 
close  enough  with  the  first  approximation.      It  deserves  to 
be  employed  much  more  than  it  is  in  work  of  this  kind. 

The  Ordnance  Survey  levels,  however,  were  not  reduced 
by  this  method.  "  The  discrepancies  in  the  levelling  along 
the  different  lines  or  routes  brought  out  in  closing  on 

49 


380  THE  ADJUSTMENT   OF   OBSERVATIONS. 

common  points  have  been  treated  for  England  and  Wales 
as  a  whole  and  rigorously  worked  out  by  the  method  of 
least  squares,  involving  ultimately  the  solution  of  a  system 
of  equations  with  91  unknown  quantities."* 

Ex.  In  the  figure  A  WMG,  A  is  the  initial  point,  height  zero,  and  the 
measured  heights  and  distances  are  as  follows  : 

A W  m 

A  1^=42.65101  Z>i  =37.8  kil. 

AM  •=  54. 74663  Z>2  =  35.8    " 

AG    =58.56223  Z>3  =  22.6    " 


Fig. 51 

WM  =  12. 10530  Z>4=44.2    ' 

MG   =    3.79892  Z>5  =  27.9    " 

required  the  adjusted  values  oi  the  heights. 

First  Solution.  —  Let  the  most  probable  corrections  to  the  five  measured 
differences  of  height  in  order  be  v\,  z/2,  z>a,  z>4,  7'5.  The  points  IV,  M,  G  are 
completely  determined  by  A  W,  AM,  AG.  We  may,  therefore,  take  v\,  z>i,  v3 
as  independent  unknowns. 

The  observation  equations  are 

v\  -=v\  weight  26.5 

Vy  =  v<2  "      27.9 

z/3  =  z/3  44-2 

z>4  =  —  vi  +  vi  —  0.00968  "      22.6 

z»5  =  —  Vi  +  z>s  +  0.01668  "       35.8 

1000 
the  weights  being  computed  from  — — - . 

The  solution  is  finished  as  in  first  solution,  Art.  140. 

Second  Solution. — From  the  closed  circuits  A  JVM,  AMG  we  have  the 
condition  equations 

v\  —  V1-\-V^=.  —  0.00968 
Vi  —  va  +  Vi,  =  +  O.OI668 
with 


J>\  = 

The  solution  is  finished  as  in  second  solution,  Art.  140. 

*  Abstract  of  Levelling  in  England  and  Wales,     Introduction,  p.  vi. 


APPLICATION   TO    LEVELLING. 


381 


ThirJ  Solution. — Adjust  each  simple  closed  circuit  in  the  figure  in  order. 
Since  in  the  circuit  AMG  the  algebraic  sum  of  the  three  differences  of  height 
should  be  zero,  we  may  apply  the  principles  of  Arts,  no,  in.  We  have 

AM=       54-74663         D*  =  35-3        .'./.,  =  27.9 
MG  =         3-79892        £>5  =  27.9  pb  =  35.8 

AC  =  —  58.56223         D3  —  22.6  /3  =  44.2 


—    0.01668 


86.3 


Correction  to  AM  =.  ~  —  X  0.01668  =  -{-0.00692 
06.3 

T 

Weight  of  adjusted  AM  =  27.9  -f- 


— 

35.8 


=  47-7 
Hence  for  the  circuit  AMW 


^^=42.65101 

=  I2.I053O 


— 

44.2 


*  =47-7 

!     =26.5 

^    =22.6 


which  may  be  adjusted  as  above. 

The  circuit  AMG  may  be  again  adjusted,  and  so  on. 

Fourth  Solution. 


I. 

Weight. 

A 

[K 

M 

G 

Means. 

26.5 
27-9 
44.2 

22.6 
358 

o. 

0. 

o. 

42.65101 

o. 

=4.74663 
12.10530 

=;8.  56223 
3.79892 

II. 

o. 

0. 
0. 

42.65101 
42.65101 

M-7-4663 

54-75631 
54.75096 

58.  56223 

<&M9&8 

Means 

o. 

42.65101 

54-75096 

58.55670 

III. 

0. 

o 
o. 

0. 

0. 

+  0.00433 

-  o  oo=;35 
o. 

-  o.oo;!;3 
+  0.00682 

0. 

»  0.00216 
-  0.00276 
-  0.00268 
+  0.00341 

IV. 

o. 
•f  o  00216 
-  0.00276 

42.65101 

42.64833 

54  74879 

M.75363 
54-75437 

58-55947 
58.55329 

Means 
Final  values 

-  o  00063 
o. 

42.64978 
42.65041 

54-75237 
54.75300 

<^.  55670 
S8-M733 

382  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Trigonometrical  Levelling. 

175.  In  extensive  surveys  in  which  a  primary  triangula- 
tion  is  carried  on,  the  heights  of  the  stations  occupied  may 
be  conveniently  found  by  trigonometrical  levelling.  The 
extra  labor  required  to  measure  the  necessary  vertical 
angles  while  the  horizontal  angles  are  being  read  at  the 
several  stations  is  but  slight. 

When  a  country  is  hilly,  heights  can  be  perhaps  as 
closely  determined  by  trigonometrical  levelling  as  by  spirit 
levelling  ;  but  if  the  country  is  flat  and  the  triangulation 
stations  low,  experience  has  shown  that  much  more  de- 
pendence is  to  be  placed  on  the  latter  method.  See,  for 
comparisons  of  the  relative  accuracy  of  the  two  methods, 
Report  of  U.  S.  Coast  Snrvty,  1876,  App.  i6and  17  ;  Report  of 
G.  T.  Survey  of  India,  vol.  ii. ;  Report  of  New  York  State 
Survey,  1882.  Sometimes,  indeed,  the  agreement  is  so  close 
that  it  must  be  regarded  as  accidental.  Thus  in  the  de- 
termination of  the  axis  of  the  St.  Gothard  Tunnel  Engineer 
Koppe  found  the  difference  of  height  of  the  two  ends  by 
trigonometrical  levelling  to  be  39"*.  13.  The  spirit  levelling 
of  precision  executed  by  Hirsch  and  Plantamour  gave  it 
39".  05. 

The  great  cause  of  inaccurate  work  in  trigonometrical 
levelling  is  atmospheric  refraction.  This  is  one  of  those 
disturbing  causes  which  is  so  erratic  in  its  character  that 
no  method  has  yet  been  devised  for  determining  it  that  is 
very  satisfactory.  Hence  the  plan  adopted  in  trigono- 
metrical levelling  is  to  observe  only  at  or  near  the  time  of 
minimum  refraction.  Without  this  precaution  very  dis- 
crepant results  may  be  looked  for.  For  example,  in  India, 
where  in  certain  districts  the  triangulation  has  been  carried 
for  hundreds  of  miles  over  a  level  country  with  stations  10 
or  12  miles  apart  and  from  18  to  24  feet  high — just  high 
enough  to  be  mutually  visible  at  the  time  of  minimum  re- 
fraction— "  numerous  instances  are  recorded  of  the  vertical 
angles  varying  through  a  range  of  6  to  9  minutes,  corre- 


APPLICATION    TO    LEVELLING. 


383 


spending   to  an  apparent  change   in   altitude  of    100  to  150 
feet  in  the  course  of  24  hours." 

176.  To  Find  the  Zenith  Distance  of  a  Signal  —  Point  the 
telescope  at  a   mark  on  the  signal  and  bisect  it  with  t he- 
horizontal    thread.      Then   turn   the   telescope    180°   on  its 
axis,  transit  it,  and   bisect  the  mark  again.     At  both  bisec- 
tions  read   the  vertical  circle  and  the  level  parallel  to  the 
vernier  arm.     One-half  of  the  difference  of  the  circle  read- 
ings corrected  for  level  will  give  a  single  determination  of 
the  value   of  the  zenith   angle  sought.     The  result  is  free 
from  index  error  of  the  circle. 

Errors  of  graduation  with  good  instruments  are  so  small 
in  comparison  with  the  uncertainties  arising  from  refrac- 
tion that  it  is  unnecessary  to  eliminate  them  in  the  measure 
of  vertical  angles. 

177.  Let  us  consider  the  general  case  in  which  the  zenith 
angles  measured  at  two  stations  whose  difference  of  height 
is  required  are  not  read  simul- 
taneously.      In    the    figure,    if 

A^  Ay  denote  the  positions  of 
the  instruments  employed  on 
two  lofty  stations,  the  angles 
that  the  observer  has  attempt- 
ed to  measure  are  Z^AVA^  and 
Z^A^A^.  But  on  account  of  the 
refractive  power  of  the  atmos- 
phere the  path  of  a  ray  of  light 
from  At  to  A,,  will  not  be  a 
straight  line  but  a  curve  more 
or  less  irregular,  and  the  direc- 
tion in  which  A^  is  seen  from  A,  will  be  that  of  the  tangent 
A^T  to  this  curve  at  A,.  The  line  of  sight  from  A.2  (oAt 
will  not  necessarilv  be  over  the  same  curve. 

The  angles  between  the  apparent  directions  of  the  rays 
of  light  at  A^A^  and  the  real  directions  are  called  refraction 


angles. 


Thus  TAtA,  is  the  refraction  angle  at  A,. 


Mem.  Roy.  Astron.  Sac.,  vol.  xxxiii.  p.  104. 


384  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Let  s,,  £„  denote  the  measured  zenith  angles  reduced  to 
the  heights  of  the  instruments  at  Alt  Av 

D=the  distance  between  the  instruments. 
C '  =  the  angle  at  the  earth's  centre  subtended  by  D. 
£,,£,,=  the  refraction  angles  at  A:,  Ay. 

Now,  assuming  the  paths  of  the  rays  of  light  from  one  sta- 
tion to  the  other  to  be  arcs  of  circles,  which  is  approximately 
the  case  when  the  lines  are  of  moderate  length,  we  have 

_  t  path  of  ray 

C —  ^~       7>       : 
Rc 

where  Rc  is  the  radius  of  the  refraction  curve. 
But  approximately 

r  _  path  of  ray 
~~R~~ 

where  R  is  the  mean  radius  of  the  earth. 
Hence  we  may  put 

C.  =  iV  C,  =  i*,C  (0 

where  k^  k^  are  constants  and  may  be  called  refraction 
factors. 

Observations  are  frequently  made  so  as  to  be  simul- 
taneous at  At,  A^  and  the  line  of  sight  may  then  be  as- 
sumed to  be  the  same  arc  of  a  circle  for  both  directions. 
In  this  case  if  we  put 

r  .\-r  —  IC 

(si   TSa  —  K^ 

k  is  called  the  coefficient  of  refraction. 

178.  To  Find  the  Refraction  Factors. — The  triangle 
CA,A^  gives  the  relation 


But,  with  sufficient  accuracy,  C=— — : Tl,  where  R  is  the 

R  sin  i 

radius  of  curvature  of  the  line  in  question. 


APPLICATION   TO    LEVELLING.  385 

Hence,  attending  to  equations  I,  Art.  177,  we  may  write 
the  above  relation  in  the  form 


*1  +  ^  =  2|r-     -^-(^+  ~2  -  1 80°)  |         (2) 

This  equation  shows  that  a  single  line  will  not  give  the 
refraction  factors,  and  we  must,  therefore,  have  a  net  of 
lines  with  the  zenith  angles  read  at  the  ends  of  each  line. 
If,  for  simplicity,  we  consider  a  quadrilateral  ABCD  we  shall 
have  six  equations  of  the  form  (2).  As  Fig.53 

these  equations  contain  twelve  unknowns 
we  may,  in  order  to  reduce  this  num- 
ber, assume,  if  the  observations  at  sta- 
tion A  over  the  lines  AB,  AC,  AD  are 
nearly  simultaneous,  that  the  k  for  each 
of  these  lines — that  is,  the  k  for  station  A 
—is  the  same.  Similarly  at  stations  B,  C,  D,  so  that  in  the 
most  favorable  case  we  shall  have  4  unknowns  and  6  equa- 
tions. 

It  may  not  always  be  possible  to  get  one  k  for  each  sta- 
tion, but  in  a  net,  if  all  of  the  lines  have  been  sighted  over, 
a  sufficient  excess  in  number  of  equations  over  unknowns 
will  in  general  be  found  to  admit  of  solution  by  the  method 
of  least  squares.  As  regards  the  weights  to  be  assigned  to 
these  equations,  we  may  proceed  in  the  ordinary  way.  If 
,7,  were  observed  ;/t  times,  and  zn  were  observed  w2  times, 
the  weights  of  .cr,  and  22  may  be  taken  to  be  w,  and  «2  re- 
spectively, and  the  first  equation  would  have  a  weight  P 
given  by 


that  is,  its  weight  would  be  proportional  to 


»« 


",  +  », 


386  THE   ADJUSTMENT   OF   OBSERVATIONS. 

179.  To  Find  the  Mean  Coefficient  of  Refraction. 

—When  the  zenith  distances  are  simultaneous, 

kl  =  £,  =  k  suppose 

and  the  refraction  coefficient  at  the  moment  of  observation 
is  for  both  stations, 


,. 

A  value  of  k  can  thus  be  found  for  each  line  sighted  over 
from  both  ends  simultaneously.  To  get  an  average  value 
of  k  for  the  whole  system  the  weighted  mean  of  these 
separate  values  must  be  taken.  The  same  method  of 
assigning  weights  may  be  used  as  in  the  preceding.  Bessel 
argues  that  errors  arising  from  irregularities  in  k  are  of 
much  more  importance  than  the  errors  in  the  zenith  angles, 
and  proposes  the  empirical  formula* 


for  assigning  relative  weights.  In  this  he  is  followed  by 
the  Coast  Survey  in  their  determination  of  the  coefficient 
of  refraction  from  observations  made  in  Northern  Georgia 
near  Atlanta  base. 

The  mean  values  of  k  found  by  the  Coast  Survey  in  N. 
Georgia  (1873)  and  by  the  New  York  State  Survey  (1882) 
were  0.143  and  0.146  respectively. 

1 80.  To  Find  the  Differences  of  Height.— There 
are  three  cases  to  be  considered: 

(i)  When  the  zenith  angles  at  both  ends  of  each  line  are 
used. 

(a)  When  the  zenith  angles  are  not  simultaneous. 

Let  //,  —  known  height  of  first  station 
//„  =  height  of  next  station 

*  Gradinessung  in  Ostfreusscn,  p.  ig6. 


APPLICATION   TO   LEVELLING.  387 

Krom  the  triangle  CA,At  (Fig.  52) 

H.-ffi  CA.  -  CA,        tanKAt-AJ      (} 


Substituting  for  At,  Ay  their  values  in  terms  of  £„  zv  £„  kv 
and  reducing, 


(b)  When  the  zenith  angles  are  simultaneous  k^-=kv 
and 


which  is  the  form  used  on  the  Coast  Survey. 

(2)  When  the  angles  observed  at  each  end  of  a  line  are 
used  separately. 

In  the  common  case  of  a  line  sighted  over  from  one  end 
only,  we  have,  since 


by  substituting  these  values  in  (i)  and  reducing, 


=  Z?  cot  ^  +  Z)2  +  -=&  cof  ^  (4) 

which  is  equivalent  to  the  Coast  Survey  form. 

When  the  line  is  sighted  over  from  both  ends  we  have 
two  values  of  the  difference  of  height,  whose  weighted  mean 
gives  the  required  result. 

50 


388  THE   ADJUSTMENT   OF   OBSERVATIONS. 

181.  Precision  of  the  Formulas  for  Differenced  of 
Height.  —  It  //„  //2,  //3  denote  the  differences  of  height  be- 
tween two  stations  found  from  non-simultaneous  readings 
of  angles  at  the  stations,  from  simultaneous  readings,  and 
from  angles  read  at  one  of  the  stations  only,  respectively, 
then,  with  sufficient  accuracy, 


7z2  =  /Man  i<>2  -  O  (i) 

£,  =  Z?  cot  #,  +  (i  -£,)£! 

Let  the  m.  s.  e.  of  an  observed  zenith  angle  be  /jtg.  De- 
note by  ///&,,  /^2,  ////3  the  m.  s.  e.  of  the  differences  of  height 
found,  and  by  fik  the  m.  s.  e.  of  a  refraction  factor  k.  Then 
by  differentiation,  taking  D,  r,,  52,  k^  /£2  as  independent 
variables,  and  remembering  that  z»  z^  are  each  equal  to  90° 
nearly,  and  that  we  may  put  dD  =  o,  since  the  distances 
are  well  known  in  comparison  with  the  heights,  we  shall 
have  (Art.  65) 


(2) 


These  results  show  that  differences  of  height  are  found 
with  the  greatest  precision  from  simultaneous  observations. 
182.  Adjustment  of  a  Net  of  Trigonometric 
Levels.  —  It  will  be  sufficient  to  consider  a  simple  closed 
figure,  as  a  net  of  levels  can  be  broken  up  into  a  number  of 
closed  figures,  generally  triangles.  For  simplicity  take  a 
triangle. 


APPLICATION   TO    LEVELLING.  389 

The  same  value  of  k  is  assumed  for  all  of  the  lines 
radiating'  from  a  station.  Denote  the  values  of  k  at  the 
three  stations  by  i\,  kv  kt  respectively.  Then  we  have  the 
observation  equations 

(          R  sin  i"  ,  } 

^=2  I  i  -      —y:  —  (*»  +  #,—•  180  )  |-    =  /,,  a  known  quan. 


The    condition    to   be   satisfied    amon^    the   differences    of 

o 

height  at  the  vertices  of  the  triangle  is  that,  on  starting 
from  any  station  and  returning1  to  it  through  the  other  two 
stations,  we  should  find  the  original  height.  Thus  proceed- 
ing round  the  triangle  in  order  of  azimuth,  if//,,  //2,  //3  de- 
note the  differences  of  height  of  the  stations,  and  (//,),  (//2), 
(//3)  the  most  probable  corrections  to  these  values,  we  must 
have 


that  is,  we  must  have  a  condition  equation  of  the  form 

ak^  -j-  bk^  -j-  ck^  —  /4 

where  a,  b,  c,  /4  are  constants. 

The  four  equations  may  be  solved  by  the  method  of  cor- 
relates, and  the  differences  of  height  may  next  be  computed 
from  equations  i,  Art.  181,  and  will  be  found  consistent. 

If  the  circuit,  instead  of  being  a  simple  one,  has  diago- 
nals, then,  as  in  Art.  173,  if  /  is  the  number  of  lines  read 
over,  and  ^  the  number  of  stations  in  the  circuit,  the  num- 
ber of  conditions  to  be  satisfied  among  the  differences  of 
height  is 

l-s+i 


390 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


Ex.  In   the   triangulation   of    Georgia,    near   Atlanta   Base,   there   were 
measured  zenith  angles  as  follows  :* 


Stations  occupied. 

Station  read  to. 

Zenith  Angle. 

Time. 

S.  W.  Base, 

(  Sweat  Mt. 
(  Stone  Mt. 

89°  37'  53"-7 
89°  26'  2i".6 

2  days. 

N.  E.  Base, 

(  Sweat  Mt. 
(  Stone  Mt. 

89°  39'  5  1  "-7 
89°  24'  04".  o 

I     " 

Stone  Mt., 

fN.  E.  Base. 
•{  S.  W.  Base. 

90°  43'  20".  8 
90°  41'  56".  8 

2       " 

[Sweat  Mt. 

90°  08'  58".  9 

Sweat  Mt., 

(S.  W.  Base. 

•I  N.  E.  Base. 
I 

9°°  33'  37"-2 
9°°  3i'  43"-4 

I     " 

[Stone  Mt. 

90°  09'  37".  o 

The  mean  latitude  is  34°  N.  approx. 


Fig.54 


N.E.B. 


Call  ki, 
respectively. 


S.W.B: 


Distances. 

S.  W.  Base-Sweat  Mt., 

N.  E.  Base-Sweat  Mt., 

N.  E.  Base-Stone  Mt.,     16321 

Stone  Mt.-S.  W.  Base,     17761 

Sweat  Mt.  -Stone  Mt.,       40726 

,  k*  the  refraction  factors  for  all  lines  at  the  stations  i,  2,  3,  4 

*  The  measures  are  taken  from  C.  S.  Report,  1876. 


APPLICATION   TO    LEVELLING.  391 

We  must  form  two  classes  of  equations: 

(i)  Observation  equations.     These  are  computed  from  the  form 


'  =  2Ji  --~ 


BD 

where  B  =  — — : -„ 

J?  sm  i 

The  weights  are  computed  according  to  the  formula  —       -  V D  (Art.  179). 

Ij  will  be  close  enough  to  take  «i,  «3  as  the  number  of  days  of  observation. 
That  is  the  best  we  can  do  with  our  data.     We   have   then   the   observation 

equations 

£1  +  £3  =0.249     weight  1.04 

£1  +  £4  =  0.267  "  1.33 

£3  +  £4  =  0.308  "  1.35 

£a  +  £3  =  0.297  "  0.80 

£2  +  £4  =0.317  "  0.85 

(2)  Condition  equations. 

The  number  of  lines       =  5 
The  number  of  stations  =  4 
.-.  The  number  of  condition  equations  =  5  —  4  +  1 

—  2 

These  two  condition  equations  arise  from  the  sums  of  two  sets  of  three  equa- 
tions, each  of  the  form 

H-  H'  =  Z>tan  \(z-z')  +  ^  (£-£') 
They  are,  for  the  triangles  134,  234, 

o  =  —  1.52  —  10.92  ki  —  4 1.79  £3  +  52.71  £4 
o=  +  1.84  +  14. 47  £2  +40.17  £3  —  54.64/^4 

The  solution  of  these  equations,  subject  to  the  relation 

[££]  =  a  minimum, 

gives 

£1=0. 1 18  £3  =  0.149 

£4  =  0.107  £4  =  0.171 

whence  the  adjusted  differences  of  height  follow  at  once  : 

m 

1  to  3        +  198.24 

3  to  4  2.30  }-  check  sum      o.oo 

4  to  i        —  195.94 

3  to  2       —  191.17 

2  to  4        +  188.88  }•  check  sum  +  o.oi 

4  to  3        +      2.30 

The  precision   of  the  adjusted  values,    or  of  any  function  of  them,  may  be 
found  exactly  as  in  Art.  114. 


392  THE    ADJUSTMENT   OF   OBSERVATIONS. 

183.  Approximate  Method  of  Adjusting  a  Net.— 

On  account  of  the  many  uncertainties  attendant  on  finding 
the  refraction  factors,  it  is  not  often  that  so  elaborate  a 
method  of  adjusting  the  heights  as  the  preceding  is  fol- 
lowed. 

In  the  ordinary  method  of  observation,  where  the  ob- 
served zenith  angles  are  simultaneous  at  every  two  stations, 

//  =  H,  —  Hl=  D  tan  |  (z,  -  z,} 

and  the  differences  of  height  may  be  computed  at  once  with- 
out any  reference  to  the  coefficient  of  refraction.  These 
differences  of  height,  considered  as  observed  quantities, 
may  be  adjusted  for  conditions  of  closure  in  the  net,  as  in 
spirit  levelling,  Art.  174. 

The  weights  P  to  be  assigned  to  the  differences  of  height 
in  the  solution  will  be  found  from 


and  therefore  the  weights  are  inversely  proportional  to  the 
squares  of  the  distances  between  the  stations. 

When  the  zenith  angles  are  not  simultaneous,  after  find- 
ing the  refraction  factors,  as  in  Art.  182,  and  computing 
the  differences  of  height,  we  should  find  the  weights  of 
these  differences  of  height  from 


It  would  seem  safe  to  assume  pz  =  2",  p.k  =  0.02. 
Now,  £Z>2sin2 1"/**  <  |  ~R*  P* 

as  D  >  4  miles. 

Hence  for  distances  between  the  stations  up  to  4  miles  the 
first  term   is  the  important  one,  and  for  greater  distances 


APPLICATION  TO   LEVELLING. 


393 


the  second  term.  We  should,  therefore,  for  distances  be- 
tween stations  not  greater  than  4  miles,  weight  inversely 
as  the  square  of  the  distance,  and  for  distances  over  that 
amount  inversely  as  the  fourth  powers  of  the  distances. 


Ex.  i.  In  the  determination  of  the  axis  of  the  St.  Gothard  Tunnel  (Fig. 
37)  the  heights  of  the  trigonometrical  stations  were  determined  by  trigono- 
metrical levelling.  The  following  were  the  results  unadjusted  with  their 
weights.  Required  the  values  adjusted  for  closure  of  circuits. 

DifT.  of  height.    Wt. 


Diff.  of  height, 

,    Wt. 

m 

Airolo-  XII. 

914.96 

23 

Airolo-       X. 

1287.75 

17 

Airolo-    XL 

1299.27 

2 

Airolo-     IX. 

1553.09 

5 

XII.-     X. 

372.73 

5 

XII.-     XI. 

384.41 

2 

XII.-     IX. 

638.30 

3 

XII.-  VIII. 

814.35 

i 

X.-    XI. 

II.  60 

3 

X.-    IX. 

265.48 

6 

X.-VIII. 

441.10 

2 

XL-    IX. 

253.87 

I 

XI.-  VIII. 

429.55 

10 

IX.-VIII. 

175-37 

I 

III.-    IX. 

216.46 

i 

V.-    IX. 

899.87 

i 

V.-VIII. 

1075-77 

i 

III.-VIII. 

391.74 

i 

VII.-VIIL 

901.78 

i 

V.-  VII. 

174-45 

7 

IV.-    III. 

296.69 

60 

VII.-    III. 

509.49 

4 

GOschenen-    III. 

1376.19 

14 

V.-    IV. 

387-24 

20 

VII.-    IV. 

212.75 

7 

Goschenen-    IV. 

1079.50 

30 

Goschenen-      V. 

692.35 

15 

The  heights  of  Airolo  and  Goschenen  are  1147.12  and  1108.07  respec- 
tively, as  found  by  spirit  levelling,  and  are  unaltered  in  the  adjustment. 
The  adjusted  heights  of  the  stations  will  be  found  to  be 


III. 

IV. 

V. 

VII. 

VIII. 


2484.26 

2187.57 

1800.36 

1974.78 

2876.07 


IX.  2700.35 

X.  2434.87 

XL  2446.49 

XII.  2062.08 


Ex.  2.  From 

//a  —  H\  •=.  D  tan  \(zi  —  z\] 
deduce 


//,  -H^D  tan  \(^  -  =,)  + 


**'  ' 


394  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  3.  If  a  series  of  points  connected  by  observed  zenith  angles  begin  and 
end  with  points  whose  heights  are  known,  then  the  number  of  conditions  to  be 
satisfied  among  the  differences  of  height  is 

/  —  S  +  2 

where  /  is  the  number  of  lines  read  over  and  s  the  number  of  points  in  the  net. 

Ex.  4.  If  zenith  angles  are  read  to  two  stations  from  a  station  between, 
show  that  the  difference  of  height;  //,  of  the  two  stations  will  be  found  from 

2  — 

h  =  Di  cot  22  —  D\  cot  z\ 


"2.R 

where  D\,  Z>2  are  the  distances  from  the  station  occupied. 

Also  show  that  when  D\  —  Z>2  the  precision  of  the  height  h  is  the  same  as 
if  the  observations  had  been  made  simultaneously  at  the  two  extreme  stations 
themselves. 


CHAPTER  IX. 

APPLICATION    TO     ERRORS    OF    GRADUATION    OF    LINE    MEAS- 
URES  AND   TO   CALIBRATION   OF   THERMOMETERS. 

Observations  for  the  determination  of  errors  of  gradua- 
tion of  line  measures,  and  observations  for  the  calibration 
of  thermometers,  may  be  discussed  together,  as  there  is  no 
essential  difference  in  the  method  of  reducing  them. 

184.  Line  Measures. — Let  AB  be  a  line  measure 
divided  into  n  equal  parts  as  nearly  as  may  be  at  the  points 
1,2,.  .  .11—1,  and  let  o  and  n  be  the  initial  and  terminal 
marks.  Comparisons  are  sup-  Fi  55 

posed  to  have  been  made  be-    *      1      I      j ( 

tween  AB  and  a  standard  of    °      *  n 

length,  so  that  the  distance  between  o  and  n  is  known. 
The  problem  proposed  is  to  rind  the  corrections  to  the 
intermediate  graduation  marks — that  is,  the  amounts  by 
which  the  positions  of  the  marks  should  be  changed  to  be 
in  their  true  relative  positions.  When  the  corrections  have 
been  applied  to  the  distances  01,  i  2,  .  .  .  these  distances 
should  be  all  the  same  proportional  part  of  the  entire  dis- 
tance AB. 

In  making  the  necessary  observations  two  methods  are 
in  common  use. 

(a)  Two  microscopes  furnished  with  micrometers  are 
firmly  mounted  on  a  frame  separate  from  the  support  on 
which  the  graduated  line  measure  rests.  The  zeros  of  the 
micrometers  are  placed  at  a  distance  apart  as  nearly  as 
possible  equal  to  the  distances  to  be  read.  The  marks 
o,  I  ;  1,2;  .  .  .  are  in  succession  brought  under  the  micro- 
scopes and  the  micrometers  read  in  each  position.  Each 
distance  is  thus  compared  with  the  constant  distance  be- 
51 


396  THE  ADJUSTMENT    OF   OBSERVATIONS. 

tween  the  micrometer  zeros,  which  differs  from  the  dis- 
tance of  the  true  positions  of  the  marks  by  a  fixed  but  un- 
known amount. 

Let  .*•„,  x^  .  .  .  xn  denote  the  corrections  to  the  gradua- 
tion marks  at  o,  i,  .  .  .  «.  Then  for  the  first  interval  o  I, 
if  M0,  Ml  be  the  readings  at  o,  I  and  c  be  the  unknown  con- 
stant distance  between  the  micrometer  zeros,  and  d  the  dis- 
tance between  the  corrected  positions  of  the  graduation 
marks,  we  should  have 


This  may  be  written 

#o  —  *i—  J  =  4,  (0 

where 

fol  =  M,-  Ma,  and  y  =  c-d 


Hence,  taking  four  spaces  only, 

x<>     x\  y  =  '»  \ 


(2) 


But  as  the  distance  between  the  initial  and  terminal  points 
is  known,  the  corrections  x0  and  x4  may  each  be  taken  equal 
to  zero.  Also,  since  the  equations  contain  four  unknowns 
and  are  themselves  only  four  in  number,  we  must  either 
reduce  the  number  of  unknowns  arbitrarily  or  make  addi- 
tional observations  involving  other  combinations  of  the- 
unknowns,  in  order  to  apply  the  method  of  least  squares. 
It  is  usually  more  convenient  to  solve  these  equations  by 
computing  the  corrections  to  the  intervals  01,02,.  .  .  first 
of  all,  and  then  the  corrections  to  the  positions  of  the  marks. 
Writing  zt  for  x^  —  xa,  ,?„  for  ,r2  —  x1}  .  .  .,  and  eliminating^, 


APPLICATION   TO    ERRORS   OF   LINE   MEASURES.  397 

we  have  from  the  above  equations,  supposing  the  first  space 
compared  with  all  of  the  others, 


(3)  ' 
•^l    —    Zi    =   t*  4    —    4  1 

But 


=  0  (4) 

Hence 


V»  3       I       4  4  3^0   l) 


(5) 


Also  -r,  =  ^,,  .*.,  =  .7,  -|-  -»>  •  •  •,  and  are  therefore  known. 

Ex,  In  order  to  find  the  corrections  to  the  double  decimetre  graduation 
marks  needed  to  change  the  nominal  into  exact  values  in  terms  of  the  interval 
o—i'«  on  a  Repsold  steel  metre  comparisons  were  made  as  follows:  Two 
microscopes  were  mounted  at  a  distance  of  om.2  approx.  from  each  other,  and 
readings  made  by  pointing  at  the  successive  double  decimetre  marks,  the 
microscopes  remaining  stationary  while  the  metre  was  run  under  them.  The 
order  of  reading  was  om.o  and  om.2  ;  0^.2  and  o"«.4;  .  .  .  om.S  and  im.o  ;  o»».o 
and  o'".2,  the  interval  o"«.o  and  om.2  being  read  on  at  the  beginning  and  end 
of  each  set  of  comparisons.  Twenty-four  sets  were  made  and  the  following 
results  obtained.* 

m  in  tit  m  n  in 

Interval,  0.2  to  0.4  =  o.o  to  0.2  +  2.1  ±  o.i 
0.4  to  0.6  —  o.o  to  0.2  +  2.6  ±  o.i 
0.6  to  o.S  =  o.o  to  0.2  +  0.7  ±  o.  i 

0.8  tO  I.O  =  0.0  tO  0.2  4-  2.2   ±0.1 


With  the  above  notation 


l  —  Sa  —  2.6 

zi  —34=0.7 


=  +  1.5,  and  xi  =  1.5 

=  —  O.6  JT2r=O. 


*  The  value  of  the  interval  o  —  i>*  was  found,  by  comparing  with  the  German  standard,  to  be 
-f-  JO*1.  65  t,  where  /  is  the  temperature  in  degrees  centigrade. 


398  THE   ADJUSTMENT   OF   OBSERVATIONS, 

or  in  tabular  form, 


Graduation 

om.o 

O.2 

0.4 

0.6 

0.8 

I'".O 

Correction 

OM.O 

-1.5 

-0.9 

+  0.1 

-0.7 

OM.O 

The  Precision.  —  Each  of  the  intervals  is  entangled  with  the  interval  o.o 
and  0.2,  and,  therefore,  the  p.  e.  given  are  not  independent.  From  the  mode 
of  measurement  it  is  evident  that  the  (p.  e.)2  of  o.o  to  0.2  is  half  that  of  the 
(p.  e.)2  of  each  of  the  other  intervals.  Hence  it  n  is  the  p.  e.  of  the  first  in- 
terval, and  r  the  p.  e.  of  each  of  the  others, 

r,2  =  2r2 


Hence 
and 


.-.  r2  =  0.007  ^i2  —  0.003 

r*S  =  ^5  (4  X  0.007  +  16  X  0.003) 
rxi  =  ±  oc*.o6 
Similarly  for  the  other  marks. 

From  equations  2  definite  values  of  the  corrections  may 
be  found  if  xv  and  x^  are  supposed  known.  There  is,  how- 
ever, a  greater  probability  of  eliminating  systematic  error 
if,  instead  of  spending  all  the  time  of  observation  in  the 
direct  comparison  of  the  single  intervals,  we  spend  part  of 
it  in  comparing  combinations  of  those  intervals.  Let,  then, 
as  the  best  arrangement  (see  Art.  153),  the  single  intervals 
o  i,  i  2,23,  34,  and  all  possible  combinations  of  intervals,  as 
o  2,  i  3,  2  4  ;  o  3,  i  4,  be  equally  well  compared.  The  micro- 
scope intervals  would,  of  course,  be  different  in  each  set  of 
comparisons,  being  approximately  01,  02,  03.  The  ob- 
servation equations  for  this  arrangement  are 


—  y\ 
x\  —  y\ 


=  /3 


(6) 


x^ 

X^ 


/14 

/0  4 


APPLICATION   TO   ERRORS   OF   LINE    MEASURES.          399 
But  x0  and  ;r4  are  known.     Hence  the  normal  equations 

+   4*1  —      JT3  —      *3  +      yi   +       }'3  =  —  la  1    +  /I  2   +   A  3   +  /l  4 

-  Xi  +  4X3  —     ^3  =  —  /OS  —  A  S  +  /S  »  +  /8  4 

-  Xi   —      Xl  +  4X3  -      I'*  —      }'3  =  —  /()  3  —  A  3  —  /2  3   +  /3  I 

+  4,1'l  —  A.  1  —  A  3  —  /2  3  —  /3  4          (?) 

+  JTl  -   JT3         +  3^3         =  —  A)  2  —  A  3  —  /3  I 

+  JTl  -   JT3  +  2J3  =  —  /u  3  —  A  4 

from  which  equations  the  most  probable  values  of  the  cor- 
rections may  be  found. 

The  Precision  of  the  Corrections  x»  xv  xy — The  m.  s.  e.  of 
an  observation  of  the  unit  of  weight  is  found  from  the  usual 
formula.  We  have 


9-6 

the  number  of  observations  being  9,  and  of  independent 
unknowns  6. 

The  weights  of  x^  xvx^  may  be  found  by  the  methods  of 
Chapter  IV. 

The  weight  of  x^  and  of  x3  is  V,  ar|d  the  weight  of  xti  is 
\°-.  Hence  the  m.  s.  e.  of  x:,  x^  x^  are  known. 

Ex.  Solve  by  this  method  the  example  in  Art.  187. 

(b)  The  work  of  reduction  is  much  facilitated  by  em- 
ploying an  auxiliary  scale,  CD,  divided  into  spaces  approxi- 
mately equal  to  those  of  AB,  and  whose  values  have  already 
been  found  by  comparison  with  some  standard.  If,  as 
befjpre,  we  suppose  the  single  spaces  compared,  and  also 
all  possible  combinations  of  spaces,  the  observation  equa- 
tions would  be  the  same  as  equations  6.  The  second  scale, 
however,  enables  us  to  find  each  microscope  interval  em- 
ployed. Hence  y^  y^  .  .  .  are  known,  and  may,  therefore, 
be  transposed  in  these  equations  and  added  to  the  terms 
/01,  /12,  .  .  .  The  observation  equations  thus  involve  only 


400  THE  ADJUSTMENT   OF   OBSERVATIONS. 

.fu,  .  .  .  ;t-4  as  unknowns.     Taking-  ^0  =  o,  ,r4  =  o,  the  normal 
equations  are 


—      X?  —      Xa  =  —  /o  l  +  A  2  +  A  3  +  A  4 
-      JTl  +  4^2  —      *3  =  —  /O  2  —  /I  2   +   /2  3  +  /a  4  (8) 

—      -l"l  —      #2  +  4-*°  3  =  —  /OS  —  As  —  <'a  3  +  /3  4 

from  which  ;r,,  ;tr2,  ;tr3  may  be  found. 

The  reduction  may,  however,  be  made  more  easily  by 
the  application  of  the  artifice  employed  in  Art.  156.  If  we 
consider  x0  and  ;r4  as  yet  unknown,  the  normal  equations 
may  be  written 

4-*o  —    Xi  —    Xi  —    Xa  —    X\  "=•       /o  i  +  /o  2  +  /o  3  +  A>  4 

—  X0  +  4x1  —    x<i  —    xa—    xt  =  —  /o  i  +  A  2  +  /i  3  +  l\  4 

—  X0  —      Xi+  4X3  —      Xa  —      X.i  =  —  !0  2  —  /I  2  +  /2  3  +  /2  4  (9) 

—  JTo  —      •fl  ~~      -^2  +  4-^3  —      Xl  —  —  '03  —  '13  —  '2  3  +  '3  4 

=   —  /O  4  —  A  4  —  /2  4  —  /3  4 


Adding  these  equations,  there  results 


This  was  to  be  expected,  because  we  have  left  the  initial 
and  terminal  points  free.  Hence  we  may  assume  any 
arbitrary  relation  between  the  corrections.  Let  us  assume 
as  most  convenient 

X0  +  Xi  +  Xy  +  X3   +  Xt=O  (lO) 

and  then  by  adding  this  relation  to  each  of  the  normal 
equations  we  have 

5^0  =  +  /O  1  +  /O  2  +  /O  3  +  /O  4 

5*1  =  —  lo  i  +  A  2  +  A  s  +  A  4 

5^2  =  —  /O  2  —  A  2  +  /2  3  +  /2  4  (J  J) 

5^8  —  —  '03  —  As  —  '23+  '34 
5.3:4  =  -  /0  4  -  A  4  -  /2  4  -  /3  4 

The  whole  solution  may  be  conveniently  arranged  in 
tabular  form.  The  sums  of  the  horizontal  rows  are  first 
found  and  then  placed  in  the  proper  vertical  columns,  with 


APPLICATION  TO  CALIBRATION  OF  THERMOMETERS.      401 

signs  changed.     The  vertical  columns  are  next  added  and 
divided  by  5. 


-  /„  , 

—  /0» 

~/03 

~/04 

Sum, 

-/!• 

-/li 

~/14 

Sum2 

—  /2  3 

-/24 

Sum3 

-/34 

Sum4 

—  Sumi 

—  Sum2 

—  Sum3 

—  Sum4 

5*0 

5*! 

5*2 

5*3 

5*4 

*0 

#i 

JT2 

*3 

** 

By  putting  x0  =  o,  xt  =  o  in  equations  11,  it  is  easily  seen 
that  we  have  the  same  results  for  x»  x»  x\  as  found  from 
equations  8. 

Ex.  The  intervals  0-5  mm.,  5-10  mm.,  .  .  .  25-30  mm.  on  a  steel  metre 
were  compared  with  the  interval  92-94  hundredths  on  a  standard  inch.  The 
whole  intervals  0-30  mm.  being  known  from  other  comparisons,  show  that  the 

p.  e.  of  the  interval  0-5  mm.,  if  the  p.  e.  of  a  comparison  is  r,  is  -   ^30. 
[Call  metre  intervals  s,,  S?,  .   .   .  st,  and  inch  interval  a.     Then 

j,  —a  +  At 
s-i  —  a  +  A.i 


where  A\,  //2,  .  .  .  A&  are  the  errors. 
By  addition, 


=  -*  ~  T 

where  [s]  is  a  known  quantity. 

Hence  eliminate  a  from  the  value  of  SL     ] 


185.  Calibration  of  Thermometers. — In  Fig.  55  let 
o,  i,  2,  ...  n  denote  the  graduation  marks  cut  on  the  glass 
stem,  Afi,  of  a  thermometer.  The  point  A  we  will  take  to 
be  the  freezing  point  and  B  the  boiling  point.  These  points 
are  known,  being  first  determined  bv  the  maker  of  the  in- 


402  THE   ADJUSTMENT   OF   OBSERVATIONS. 

strument,  and  can  be  redetermined  at  any  time  by  special 
experiments. 

We  will  suppose  that  the  errors  of  the  freezing  and  boil- 
ing" point  marks  are  known,  and  proceed  to  consider  the 
corrections  to  the  intermediate  marks  due  to  want  of 
uniformity  of  the  bore  of  the  stem,  or  the  calibration  correc- 
tions, as  they  are  called.  If  the  bore  between  two  assigned 
marks  were  uniform  it  could  be  filled  by  a  certain  volume 
of  mercury.  The  length  of  this  standard  volume  or  column 
would  be  indicated  by  the  readings  at  the  two  marks.  If 
the  column  were  moved  along  until  it  came  between  two 
other  marks  with  the  same  difference  of  readings  as  before, 
and  the  bore  were  not  uniform  with  the  bore  at  first  posi- 
tion, the  column  would  not  fill  the  stem  between  the  marks. 
The  amount  of  difference  would  be  the  calibration  correc- 
tion for  this  interval. 

In  explaining  the  method  of  determining  the  calibration 
corrections  let  us  for  simplicity  consider  3  intermediate 
points  only  between  the  freezing  and  boiling  points;  that 
is,  4  intervals,  o  i,  I  2,  2  3,  3  4,  of  45°  each  on  Fahrenheit's 
scale.  A  column  of  mercury,  of  volume  sufficient  to  fill  o  I 
as  nearly  as  may  be,  is  broken  off  and  the  ends  read  when  in 
the  positions  o  i,  I  2,  2  3,  3  4.  Another,  equal  to  o  2,  is  read 
in  the  positions  02,  i  3,  2  4,  and  a  third,  equal  to  o  3,  in  the 
positions  03,1  4.* 

As  it  is  impossible  to  break  off  the  exact  column  re- 
quired in  every  case,  we  break  off  one  as  nearly  equal  to  it 
as  possible  and  neglect  the  error  introduced  by  the  small 
discrepancy,  which  need  not  exceed  o°.2. 

1 86.  The  following  method  of  breaking  off  column 
lengths  was  that  employed  by  Mr.  Charles  C.  Brown,  of  the 
Lake  Survey,  in  determining  the  calibration  corrections  of 
thermometer  5280  Green  (New  York): 

A  column  of  mercury  200°  to  250°  in  length  was  easily 
obtained  by  making  that  amount  of  mercury  run  into  the 

*  This  method  of  making  the  readings  is  known  as  Neumann's.     See  the  corresponding  form 
in  triangulation  Art.  154. 


APPLICATION  TO  CALIBRATION  OF  THERMOMETERS.      403 

stem,  a  fe\v  slight  jars  being  sufficient  to  start  the  column 
moving;    a    little    manipulation    then    brought    the    empty 
space  in  the  bulb  to  the  junction   of  the  bulb  and  the  stem, 
when  a  sudden  turn  of  the  thermometer  upright  broke  off 
the  column,  and  almost  as  sudden  an  inversion  preserved  it. 
When  too  large  a  space  was  left  in  the  bulb  to  be  filled  by 
heating  the  thermometer  to   140°   or   150°,  a   few   drops  of 
mercury  were  allowed  to  drop  off  the  end  of  the  column  in 
the  stem  held  upright,  good  care  being  taken  to  stop  the 
operation  before  the  column  joined  the  mercury  in  the  bulb. 
Then,  the  thermometer  being  heated  until  the  mercurv  from 
the  bulb  began  to  appear  in  the  stem,  the  column  already  in 
the  stem  was  run  down  carefully,  and  partially  joined  to  the 
mercury  in  the  bulb,  leaving  a  small  bubble  on  one  side  of 
the  column,  the  thermometer  being  allowed   to  cool  slowly 
until  the  desired  length  of  column  above  this  bubble  (which 
remains  very  nearly  stationary)  was  obtained.      The  column 
was  broken  off  at  the   bubble  by  a  slight  twitch  or  jar.     If 
there  are  objections  to   heating  the  thermometer  above  a 
certain  temperature,  column  lengths  above  10°  to  20°  or  3OD 
longer  than  the  number  of  degrees  of  that  temperature,  de- 
pending on  the  distance  of  the  32°  point  from  the  bulb,  can 
be  obtained  by  jarring  off  small  drops  of  mercury  from  a 
long  column   into   the  reservoir  at  the  top  of  the  stem.     It 
requires  much   more  time,   care,  and  patience  to  obtain  a 
column  in  this  way  than  in  the  other.     Columns  more  than 
about   160°  in  length  were  so  obtained   in   this   cnse.      It  is 
rather  difficult  to  break  off  short  columns  5°  to  15°  in  length 
in  the  manner  first  described,  the  weight  of  mercury  in  the 
short  column   not  giving    momentum   enough    to    move  it 
away  from  the  rest  of  the  column  readily.     A  little  patience 
is  all  that  is   necessary,  however.     To  be  able  to  read  the 
shorter  columns  at  32°  a  column  10°  to  50°  long,  depending 
on  the  temperature  at  the  time,  must  be  broken  off  and  put 
into  the  reservoir  at  the  upper*end  of  the  tube,  out  of  the 
way. 

187.    Let  xv,  -i',,  .r3,  ,r3,  x^  denote  the  calibration  correc- 
52 


404  THE   ADJUSTMENT   OF   OBSERVATIONS. 

tions  at  the  several  graduation  marks.  Then,  approxi- 
mately, we  have  for  each  interval  a  relation  of  the  form 

column  —  diff.  of  read  ings  -(-  diff.  ofcal.  corr. 

As  the  volumes  of  the  columns  broken  off  correspond  to  the 
constant  interval  between  the  microscopes  in  the  case  of 
line  measures,  the  observation  equations  furnished  by  the 
thermometer  readings  will  correspond  in  form  to  equations 
6,  Art.  184. 

The  rigorous  solution  of  these  equations  is,  however, 
rather  complicated,  and  though  in  comparisons  of  stand- 
ards of  length  it  may  be  allowable,  from  the  precision  with 
which  measurements  can  be  made,  to  spend  the  labor  de- 
manded by  this  form  of  reduction,  yet  in  thermometric  work, 
where  the  readings  cannot  be  very  close  from  the  nature  of 
the  case,  an  approximate  foraris  sufficient.  By  the  following 
artifice,  which  is  quite  analogous  to  that  introduced  by 
Hansen  in  the  adjustment  of  a  triangulation  (Art.  155),  the 
labor  is  reduced  very  materially. 

Instead  of  finding  the  corrections  to  the  graduation 
marks  directly,  the  corrections  to  the  several  intervals  be- 
tween the  graduation  marks  may  first  be  found,  and  thence 
the  corrections  to  the  marks  at  the  ends  of  the  intervals. 
Thus  if  £,,  £„,  £„,  £4  denote  the  corrections  to  the  4  inte/vals 
in  our  example,  then 


f     -    Y     -    Y  "~     -    Y     -    Y  (  T  ^ 

"Z  --  't2  •*!  "4  -  -*4  -i-S  \l  J 

From  the  observation  equations  6  we  have,  by    subtract 
ing  in  pairs, 


_  r-    -  /  _  / 

""4  -      34  *a 

-   /  -    / 

—  *i  3  *0 

-  s  —  /  -  / 

-"      —   '  * 


APPLICATION  TO  CALIBRATION  OF  THERMOMETERS.       405 

Hence,  considering /ia  — /0 ,,  /33  — /)2,  ...  as  independently 
observed  quantities,  we  have  the  normal  equations 


-1  —  22  —  ;3  —  -4  — 
Z\  +  3C'l  —  23  —  24  = 
2l  —  23  +  323  —  24  = 
2,  —  2a  —  2j  +  3=4  = 


(A  a  —  /o  i)  +  (/i  3  —  /oa)  +  (A  4  -  /o  3) 

-  (A  a  -  /o  ,)  +  (A  3  -  /,  a)  +  (A  4  -  /,  3) 

-  (/a  3  -  A  a)  +  (A  4  —  /a  a)  —  (As-  /o  a) 

-  (A  4  -  A  3)  -  (A  4  -  A  3)  -  (A  4  -  /o  3) 


(3) 


which  equations  when  added  give  o  =  o  identically.  The 
reason  is  that  we  have  not  yet  fixed  the  initial  or  terminal 
points.  To  do  this  we  may,  as  most  convenient,  assume  the 
relation 

2l  +  2a    +   SS  +  Si  =  O  (4) 

Adding  this  relation  to  each  of  the  normal  equations,  we 
find  the  values  of  the  unknowns.  Thus 


4=1  =  (A  ,  -  /„ ,)  +  (/,  3  -  /« ,)  +  (/,,-  /„  3) 
423  =  -  (A  •,  -  /„  0  +  (A  3  -  A  2)  +  (A  4  -  A  3) 
4c3  =  -  (A  3  -  A  a)  +  (A  4  -  A  3)  -  (A  3  -  A.  „) 
434  =  _(/, 4  _/,,)_  (/.,  4  _  /, ,)  _  (/,  4  _  /0  3) 

The  computation  of  the  unknowns  may  be  much  facili- 
tated by  arranging  in  tabular  form,  as  follows: 


-1 

*a 

,3 

-4 

-  (A  -2  -  /O  ,) 

-(A3-/o,) 

-(A,  -A,  3) 

Sumi 

-  (A  3  -  A  a) 

-  (A  4  -  A  3) 

Surria 

-  (A  ,  -  A  3) 

Siini3 

—  Sum, 

—  Sum  2 

—  Sums 

42l 

422 

4  =  3 

4=i 

-i 

-2 

~3 

-4 

The  corrections  ,r,,  x.r  xz  are  then  found  from  the  relations 


406 


THE   ADJUSTMENT   OF   OBSERVATIONS. 


since  the  correction  x0  may  be  assumed  to  be  zero,  and  then 
Xi  must  also  be  zero  from  the  relations  (i)  and  (4). 

As  a  check  we  notice  that  the  values  of  £4  and  xa  must  be 
equal  but  of  opposite  sign. 

The  Precision.  —  If  z//,  v,',  .  .  .  denote  the  residuals  of 
equations  2,  we  have  for  the  m.  s.  e.  of  a  column  length 


6-3 


the   number  of  equations  being  6  and  of  independent  un- 
knowns 3. 

Now,  since  the  column  length  involves  the  difference  of 
the  calibration  corrections  at  its  ends,  if  we  assume  the 
m.  s.  e.  of  these  corrections  to  be  equal  we  shall  have  the 
m.  s.  e.  of  a  calibration  correction  by  dividing  the  above 

result  by    V2  ;  thus 


2(6  -  3) 

The  weights  of  the  unknowns  or  of  any  functions  of  them 
may  be  found  by  the  methods  of  Chapter  IV. 


Ex.  The  following  were  the  observed  values  of  the  lengths  of  the  45°,  90°, 
and  135°  columns  of  thermometer  Green  4470,  made  to  determine  the  calibra- 
tion corrections  at  the  77°,  122°,  and  167"  points  : 


45°  col. 

90°  col. 

135°  col. 

44°.  68 

90°.  07 

134°.  61 

44°.  7i 

90°.  09 

134°.  68 

44c-72 

90°.  1  1 

44°.  75 

The  corrections  at  the  freezing  and  boiling  points  are  known. 


APPLICATION  TO  CALIBRATION  OF  THERMOMETERS.       407 
Solution.   Arranging  in  tabular  form, 


., 

-2 

23 

.. 

Surns. 

-0-.03 

—  O°.O2 

—  o°.oy 

—  O°.I2 

—  o".oi 

—  O°.O2 

-°°-°3 

+  O'  .  1  2 

+  0-.03 

+  0°.I2 

o°.oo 

.     O°.OO 

—  0°.I2 

+  o°.03 

O°.OO 


and 


Also 


=  o°.O3 


188.  The  following  memoirs  may  be  consulted  :  Sheep- 
shanks, Monthly  Notices  Royal  Astronomical  Society,  vol.  xi. 
pp.  233-248 ;  Hanscn,  Von  der  Bestimmung  dcr  TJieilnngs- 
fcliler  eines  gradlinigcn  Maassstabcs,  Leipzig-,  1874;  Russell, 
American  Journal  of  Science,  vol.  xxi.  pp.  373-379;  Thorpe, 
Report  /-iritish  Association  for  Advancement  of  Science,  1882, 
pp.  145-204;  Brown,  Van  Ncstrand's  Engineering  Magazine, 
vol.  xxix.  pp.  1-7;  Benoit,  Mesures  de  dilatation  et  comparisons 
des  regies  mc'triqucs,  Paris,  1883. 


CHAPTER   X. 

APPLICATION  TO  EMPIRICAL  FORMULAS  AND   INTERPOLATION. 

189.  In  all  discussions  hitherto  we  have  considered  the 
observed  quantity,  whether  a  function  of  one  or  more 
variables,  to  be  a  function  whose  form  was  known  and 
which  could  therefore  be  developed  in  terms  of  that 
variable.  In  the  physical  sciences  we  meet  with  a  differ- 
ent problem.  From  observation  we  have  values  of  a  func- 
tion corresponding  to  certain  known  values  of  a  variable, 
and  are  required  to  determine  the  form  of  the  function  from 
the  observed  values;  in  other  words,  to  find  the  algebraic 
formula  connecting  the  function  and  the  variable.  The 
method  of  least  squares  will  not  enable  us  to  find  the 
most  probable  form  of  this  function.  All  it  will  do  is  to 
show  how  to  approach  more  closely  to  its  form  after  that 
form  has  been  found  approximately  by  other  means. 

For  example,  the  observed  height  of  the  tide  at  a  place 
may  be  considered  a  function  of  the  time  of  day.  If  obser- 
vations are  made  with  a  staff  gauge  three  or  four  times  a 
day,  and  the  stage  of  water  at  a  time  not  observed  is 
required,  we  cannot  interpolate  between  the  observed 
values  in  the  most  probable  manner  till  the  function  con- 
necting the  height  and  the  time  of  day  is  known. 

Sometimes,  indeed,  the  observations  may  be  so  taken  as 
to  record  themselves.  Thus  with  Saxton's  self- registering 
tide  gauge,  in  common  use  in  the  United  States,  the  curve 
representing  the  rise  and  fall  of  the  water  is  traced  continu- 
ously on  a  web  of  paper  moving  uniformly  past  a  pencil 
point  by  means  of  clock-work.  Plotting  the  time  along 
one  edge  of  the  paper  as  abscissa,  the  stage  of  water  at 


APPLICATION   TO   EMPIRICAL   FORMULAS,    ETC.  409 

any  time  required  can  be  read  at  once  from  the  sheet. 
The  curve  representing  the  rise  and  fall  of  the  water  is 
in  this  case  completely  known,  and  no  computation  is 
necessary. 

190.  From  the  nature  of  physical  phenomena,  in  which 
observations  are  made  only  at  intervals,  and  into  which  sys- 
tematic error  largely  enters,  we  must  admit  that  it  is  most 
probable  that  the  function  connecting  the  observed  quantity 
and  the  variable  is  different  for  each  observation.  We 
should  thus  have  a  series  of  equations  of  the  form 

M,  =/  (x,  a,  &,c,  . 

M,  =ft  (x,  /i,  k,  I,  . 

where  J/,,  Mt,  .  .  .  are  the  observed  values,  x  the  variable, 
and  a,  b,  .  .  .  /  //,  k,  .  .  .  are  constants. 

In  order  to  apply  the  method  of  least  squares  to  de- 
termine the  constants,  the  functions  must  first  be  reduced 
to  the  linear  form,  and  the  number  of  quantities  to  be  de- 
termined must  be  made  less  in  number,  arbitrarily  it  may  be, 
than  the  number  of  observations.  The  first  point  is  to 
determine  as  nearly  as  we  can  the  general  form  of  the 
functions  connecting  the  observed  quantities  and  the  vari- 
able. 

For  this  no  general  rule  can  be  given.  From  theoretic 
considerations  we  may  hit  upon  a  certain  formula  as 
plausible.  It  must  then  be  found  by  trial  how  closely  it 
fits  the  observed  values.  It  is  often  very  convenient  to 
make  a  graphical  representation,  the  variable  being  the 
abscissa,  and  the  observed  values  of  the  function  correspond- 
ing to  known  values  of  the  variable  the  ordinates.  The 
points  plotted  will  in  general  fall  so  irregularly,  here  a 
group  and  there  a  vacant  space,  that  a  free-hand  curve  only 
can  be  drawn  which  will  conform  to  the  general  outline  of 
the  plot.  The  curve  will  represent  the  general  form  of  the 
function  we  are  seeking.  This  method  of  eliminating  ir- 
regularity by  drawing  a  mean  curve  to  represent  the  value 


4IO  THE   ADJUSTMENT   OF   OBSERVATIONS. 

of  the  function  corresponds  to  assuming  a  common  form  for 
the  algebraic  functions  f^fv  .  .  .  above. 

A  good  example  is  afforded  by  the  method  employed  by 
Humphreys  and  Abbot  in  their  Mississippi  River  work  for 
finding  the  law  of  velocities  below  the  surface  in  a  plane 
parallel  to  the  current.*  The  observations  for  velocity 
were  made  with  floats  at  different  depths,  started  from 
boats  anchored  at  vaiious  distances  from  the  shore.  In  the 
first  place,  all  of  the  velocities  observed  from  each  anchorage 
were  plotted  on  cross-section  paper,  the  depths  of  the  floats 
being  the  ordinates  and  the  velocities  the  abscissas.  The 
resulting  curves  showed  marked  irregularities  of  form.  To 
eliminate- errors  of  observation  the  velocities  were  grouped 
in  sets  corresponding  to  nearly  equal  depths  of  water  and 
to  nearly  equal  velocities  of  the  river,  and  the  means  plotted. 
These  curves  indicated  a  law,  though  not  sufficiently  clearly 
to  allow  of  deducing  an  algebraic  expression  for  it.  "  It  is 
manifest  that  some  further  combination  is  necessary  in 
order  to  eliminate  the  effect  of  disturbing  causes.  Since 
the  absolute  depths  differ,  this  can  only  be  done  by  com- 
bining the  velocities  at  proportional  depths."  A  curve 
was.  therefore,  plotted  from  the  mean  of  all  the  velocities  at 
proportional  instead  of  at  absolute  depths.  This  curve 
proved  to  be  approximately  a  parabola  with  axis  horizontal 
and  vertex  about  o  3  of  the  depth  of  the  river  below  the 
surface.  It  could,  therefore,  be  expressed  by  an  algebraic 
formula. 

In  many  cases  we  may  find  the  form  of  the  function 
over  which  the  observations  extend  by  means  of  a  formula 
of  interpolation.  Thus  if  x'  is  an  approximate  value  of  x, 
then  for  observations  in  the  vicinity  of  x  we  have,  by 
Taylor's  theorem, 

/(*)=/{*' +  (*-*')| 

=/(*')  +  &(*  -  *'}  -f  c(*  -  *y  +  •  •  • 

where  b,  c,  .  .  .  are  unknown  constants  to  be  determined. 

*  Physics  and  Hydraulics  of  the  Mississippi  River.     Washington,  1876. 


APPLICATION   TO   EMPIRICAL   FORMULAS,  ETC.  41  I 

Having  found  the  form  of  the  function  approximately 
by  such  means  as  the  preceding,  we  can,  by  the  method 
of  least  squares,  determine  the  most  probable  values  of  the 
constants  involved,  so  far  as  the  observations  on  hand  are 
concerned.  For  let  the  function  be  reduced  to  the  linear 
form,  and  the  most  probable  values  of  the  constants  and 
the  precisions  of  these  values  may  then  be  found  by  the 
ordinary  rules  for  observation  equations  as  laid  down  in 
Chapter  IV. 

191.  The  question  naturally  arises  as  to  the  number  of 
terms  of  a  series,  such  as  the  above,  that  should  be  taken 
in  any  special  case.  Cut  and  try  is  our  main  guide.  II  the 
plot  of  the  observations  shows,  for  example,  that  the  phe- 
nomenon can  be  very  closely  represented  by  a  straight  line, 
we  should  take  the  first  two  terms.  Thus  it  has  been  usual 
to  take  the  relation  between  the  length,  V,  of  a  bar  of 
metal  and  the  temperature,  t,  to  be  of  the  form 

V—M-\-b(t-f} 

where  M  is  the  observed  length  at  the  temperature  /',  and  b 
is  the  coefficient  of  expansion  to  be  found  from  the  observa- 
tions. More  refined  methods  of  observation  indicate,  how- 
ever, that  this  expression  is  not  sufficient  to  express  the 
relation  between  length  and  temperature.  The  plot  of  the 
observed  values  indicates  a  curve  of  higher  dimensions  than 
the  first,  so  that  we  must  take  additional  terms  of  the  series 
to  represent  it,  thus  : 

V—  M+  b(t  -  t'}  +  c(t  -  tj  -f  .   .  . 

A  second  guide  is  afforded  by  the  value  obtained  of  the 
sum  of  the  squares  of  the  residuals  of  the  observation  equa- 
tions. If  the  nature  of  the  observations  is  such  as  to  war- 
rant the  expectation  of  a  value  much  smaller  than  that 
obtained,  we  may  take  additional  terms  of  the  series;  and  if 
this  is  still  unsatisfactory  we  must  seek  another  function. 
This  test  may  also  be  used  in  comparing  one  form  of  func- 
tion with  another. 
53 


412  THE   ADJUSTMENT   OF   OBSERVATIONS. 

In  anV  case,  even  when  all  of  the  tests  applied  appear 
satisfactory,  we  cannot  say  that  our  final  result  is  the  most 
probable  value  of  the  function  that  can  be  found.  We  have 
made  too  many  assumptions  for  that.  In  the  first  place, 
the  form  of  the  function  assumed  as  a  first  approximation  is 
too  uncertain;  and,  again,  \ve  have  arbitrarily  assumed  the 
number  of  constants  to  be  determined,  and  determined  this 
limited  number  so  as  to  satisfy  a  special  set  of  observations 
only.  All  that  our  final  formula  will  enable  us  to  do  is  to 
interpolate  in  the  most  probable  manner  within  the  range  of 
the  special  set  of  observed  values,  but  not  to  extrapolate. 
If,  however,  the  formula  has  been  tested  by  many  series  of 
observations  made  under  the  most  diverse  conditions,  and 
is  found  to  satisfy  them  well,  we  can  with  confidence  apply 
it  in  cases  where  no  observations  have  been  made.  We 
may,  in  fact,  consider  that  we  have  found  the  law  connecting 
the  function  and  the  variable. 

192.  It  is  evident,  too,  that  different  experimenters  may 
derive  formulas  widely  different  to  represent  like  phenome- 
(iia,  and  that  each  formula  may  satisfy  the  special  set  of 
observations  irom  which  it  was  derived  tolerably  well. 
Each  experimenter  may  have  chosen  his  constants  differ- 
ently, as  well  as  a  different  form  of  function  to  begin  with. 
As  an  example  of  this  we  may  cite  the  formulas  proposed  * 
to  represent  the  variation  of  the  elastic  forces  of  vapor  at 
different  temperatures  /. 

Young  proposed 

P  —  (a  -f  /;/)"' 

a  and  b  being  constants  to  be  found  from  observation. 
Roche, 

X 

P  =  a  a  J  f  mx 

where  x  =  /  -\-  20°. 
Biot  and  Regnault, 

log  P  =  a  -f-  bo*  +  cjf 

*  Mousson,  Fhysik,  vol.  ii.     Zurich,  1880. 


APPLICATION   TO    EMPIRICAL   FORMULAS,  ETC. 


413 


Of  these  formulas  the  first  involves  two  constants,  the 
second  three,  and  the  third  five.  The  first  curve,  there- 
fore, requires  two  observations  to  determine  it,  the  second 
three,  and  the  third  five.  If  a  group  of  observations  were 
plotted  with  temperatures  as  abscissas  and  pressures  as  ordi- 
nates,  then,  since  the  last  lormula  represents  a  curve  passing 
through  five  of  these  points,  it  is  evident  that  if  the  obser- 
vations are  good  this  curve  would  in  all  probability  pass 
near  all  intermediate  points  and  more  closely  than  a  curve 
fixed  by  only  two  or  three  points.  It  should,  therefore,  be 
chosen. 

Ex.  i.  The  following  are  the  results  of  the  observations  m;ide  for  velocity 
of  current  of  the  Mississippi  River  by  Humphreys  and  Abbot  at  Carrollton  and 
Baton  Rouge  in  1851.  Each  is  the  mean  of  222  observations  and  is  given  at 
proportional  depths,  the  whole  depth  being  represented  by  unit}*. 


Propor.  depth  of  float 
below  surface. 

Obs.  velocity  in  feet 
per  second. 

O.O 

3-I950 

O.I 

3.2299 

O.2 

3-2532 

0-3 

3.2611 

0.4 

3-25I6 

0-5 

3.2282 

0.6 

3.1807 

0.7 

3.1266 

0.8 

3-0594 

0.9 

2-9759 

In  Art.  191  it  has  been  shown  that  the  curve  of  velocities  is  approxi- 
mately parabolic  in  form.  As  a  first  approximation,  therefore,  we  take  three 
terms  of  the  expansion  from  Taylor's  theorem, 

V  =  a  +  bD  +  fD- 


where  V  \s  the  velocity,  D  the  proportional  depth,  and  a,  l>,  c  constants  to  be 
determined  from  the  observations. 


414  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Hence  the  observation  equations 

3.1950  =  0  - 

3.2299  =  a  +  o.ib  +  o.oic 
3.2532  =  0  +  .26+  .04<r 
3.2611=0+  .36  +  .ogc 
3.2516  =  0  +  .4^  +  .  i6c 
3.2282  =  0  +  .53  +  .25<r 
3.  1807  =  a  +  .63  +  .36<r 
3.1266  =  0  +  .7*5+  .49^ 
3.0594  =  0  +  .83+  .64<r 
2.9759  =  a  +  0.9^  +  O.Sl,r 

To  solve  these  equations  proceed  as  in  Art.  104.     Take  the  mean  and  sub- 
tract from  each,  and  we  have 

+  0.0188  =  —  0.45*5  —  0.2851: 
+  0.0537  =  —  °-35^  —  0-275<: 
+  0.0770=  —0.25^  —  0.245^ 
+  0.0849  =  —  o.  15$  —  o.  1951: 
+  0.0754  —  —  o.os/>  —  o.  i25c 
+  0.0520  =  +  o.os/;  —  O.O35C 
+  0.0045  =  +  o.  15^  +  0.075C 

—  0.0496  =  +O.25/;  +  0.205^ 

—  0.1168  =  +  0.35^4-  0.355^: 

—  0.2003  =  +  0.45^  +  0.5251: 

Hence  the  normal  equations 

—  0.2031  =  0.8250^  +-0.74251: 

—  0.2232  =o.7425/>  +  o. 


and  ^  =  0.4424        c=—  0.7652        «  =  3.  1952 

.'.    V—  3.1952  +  0.4424!)  —  o. 


The  residuals  r/are  found  by  substituting  for  a,  l>,  c  in  the  observation  equa- 
tions, and  give 


APPLICATION   TO   EMPIRICAL   FORMULAS,    ETC.  415 

If  we  take  four  terms  of  the  expansion,  so  that 

V  =  a  +  bD  +  cDn-  +  <//>< 
and  proceed  similarly,  we  shall  find 

a  =  3-1935,         ^=  +  0-4735         c-—  0.8563,         </— 4-  0.0675 
so  that 

V '  =  3.1935  +  0.4735  D  —  O.S563/)2  +  0.0675  D* 

and  the  sum  of  the  squares  of  the  residuals  z>  is  given  by 

[H  =0.45 

The  sum  of  the  squares  of  the  residual  errors  being  less  than  in  the  former 
case,  we  conclude  that  the  observations  are  better  represented  by  the  formula 
last  obtained. 

It  is  to  be  borne  in  mind  that  in  the  application  of  the  method  of  least 
squares  only  one  set  of  measured  quantities,  if  more  than  one  occur  in  the 
problem,  can  be  considered  variable.  The  error  is  thrown  into  this  set. 
Thus  in  the  problem  just  solved  the  depths  are  supposed  to  be  correctly 
measured  and  the  error  alone  to  occur  in  the  measured  velocities.  So  in 
finding  the  expansion  of  a  body  it  is  usual  to  consider  all  of  the  error  as 
occurring  in  the  observed  lengths,  and  to  take  the  thermometer  readings  to 
be  correct.  The  justice  of  these  assumptions  may,  however,  very  fairly  be 
questioned.  A  discussion  of  the  general  problem  covering  such  cases  will 
be  found  in  Art.  106. 


Ex.  2.  The  tension,  H,  of  saturated  stearn   at  temperature  f  C.  is  found 
from  the  formula* 


where  a  =  0.0036678,  and  a,  b,  c,  .  .  .  are  constants,  >/  +  i  in  number,  to 
be  determined  from  observed  values  of  /  and  H.  We  have  also  given  that 
when 

t  =  iooc  C.,     H  =  760  «""»• 

*  Travaujc   ft  Mtmoircs   <fu    Bureau    international  lies   raids   ct  Mf  surfs,    Paris,    1881. 


416  THE   ADJUSTMENT   OF   OBSERVATIONS. 

To  find  the  constants  a,  />,  c,  .  .  .  we  proceed  as  follows: 
Substituting  the  values  of  I  and  H  in  equation  i,  and  eliminating  a,  we 
have 


=  7OO    IO      \i  +  looo        i  -(-  at)  \i+iooo        i  -j-  °-t) 

=  760  10  ""  ^'  +  c^2  +  •  •  •)  suppose. 


log  760  —  log  //  =  bpi  +  fpv  +  .   .  .  (2) 

Next  compute  approximate  values  of  /',  t,  ...  by  selecting  «  of  the  observa- 
tions and  substituting  the  observed  values  of  t  and  H  in  Eq.  2.  Let  b' ,  c',  .  .  . 
denote  the  values  found  from  the  solution  of  the  resulting  equations,  and  let 
//'  denote  the  corresponding  value  of//,  so  that 

If '  =  760  io~^>'+c>*+-  '  •>  (3) 

Also,  let  x,  y,  .  .  .  denote  the  corrections  to  be  added  to  the  approximate 
values  b' ,  c ,  .  .  .  to  find  the  most  probable  values,  so  that 


then,  from  (2), 

log  760  —  log  \ff'  +  (^ —  ff')\  =  (b'  +  •r)/i  +  ' 
or  expanding  by  Taylor's  theorem  and  remembering  that,  from  (3), 

log  760  —  log  //'  =  b'pl  +  c'p-i  +  .    .    . 
we  have 


the  linear  form  required.     The  solution  may  be  finished   in    the   usual   way, 
as  given  in  Chapter  IV. 

193.  The  preceding  examples  show  that  as  soon  as  it  has 
been  decided  what  formula  will  apply  to  a  special  set  ot 
observations  the  main  difficulty  is  in  reducing  this  formula 
to  the  linear  form.  When  this  has  been  done  so  that  the 
formula  is  in  the  form  of  an  ordinary  observation  equation, 
the  reduction  by  the  method  of  least  squares  is  straight- 
forward. 


APPLICATION   TO    EMPIRICAL   FORMULAS,  ETC.  417 

In  cases  where  the  data  are  insufficient  or  contradictory 
it  is  useless  to  waste  time  on  long  computations,  as  the 
graphical  plot  will  show  at  a  glance  all  that  the  observations 
can  show.'  With  cross-section  paper  plots  can  be  made 
rapidly,  and  by  transferring  to  tracing  linen  one  plot  can 
be  placed  over  another,  so  that  comparisons  can  be  readily 
effected  and  a  mean  value  struck. 

In  the  problem  of  the  law  of  velocities  in  rivers,  alter 
having  decided  from  the  various  plots  that  the  curve  of 
velocities  is  approximately  parabolic  in  form,  it  is  better  to 
employ  the  method  of  least  squares  to  determine  the  con- 
stants of  the  curve  rather  than  to  trust  to  the  graphical 
method  throughout,  but  only  for  the  reason  that  the  data 
available  are  very  complete. 

194.  Periodic  Plienoinena. — A  large  class  of  physical  phe- 
nomena is  more  or  less  periodic  in  character.  The  daily 
temperature  throughout  the  year  at  a  given  place,  the  er- 
ror of  graduation  of  the  limb  of  a  theodolite,  the  magnetic 
declination  with  reference  to  the  time,  etc.,  are  examples. 
The  phenomenon  may  not  be  strictly  periodic  in  that  like 
periods  succeed  each  other  in  their  proper  order,  or  that 
even  any  one  period  is  perfect  throughout.  If  a  plot  of  the 
observed  values  of  the  function  corresponding  to  certain 
values  of  the  variable  involved  be  made,  and  it  indicates  the 
periodic  character  of  the  function,  we  may  assume  as  the 
form  of  the  function  a  number  of  terms  of  the  series  fur- 
nished by  Fourier's  theorem.  Each  observation  will  give 
an  observation  equation,  and  from  the  observation  equations 
the  values  of  the  constants  in  the  formula  will  be  deter- 
mined. As  in  the  cases  already  discussed,  the  final  formula 
is  to  be  looked  on  as  holding  only  within  the  limits  of  the 
observations,  serving  as  a  guide  for  the  future  study  of  the 
phenomenon  in  question,  and  only  to  be  used  with  great 
caution  outside  the  limits  of  the  observed  values. 

Suppose,  then,  that  n  observations  give  the  values 
yl/,,  J/a,  .  .  .  Mn  corresponding  to  the  values  <p,  <p-\-  6,  .  .  . 
<f  -j-  (n  —  i)#  of  the  variable  <f  uniformly  distributed  over 


41 8  THE   ADJUSTMENT   OF   OBSERVATIONS. 

the  period  represented  by  360° ;  that  is,  such  that  the  inter- 

i  &  •    360° 
val  6  is  -—. 

n 

Now,  if  f(<p]  is  any  arbitrary  periodic  function  of  a  vari- 
able <p,  we  may,  according  to  Fourier's  theorem,  write 

f(<p)  =  X+  h,  sin  (a,  -f  <p )  -f  /tt  sin  (a,  +  2<p)  +  .   .   . 

where  X,  //,,  hv  .  .  .  «1?  «2,  .  .  .  are  constants  to  be  deter- 
mined from  the  observed  values  of  f(<f>)  corresponding  to 
assigned  values  of  the  variable  <p.  Substitute  Mlt  M^, 
M3,  .  .  .Mufor/(<p),  and  <p,  <p -f  6,  <p  +  26,  .  .  .  <p  +  (;/  -  i)0 
for  <f>,  in  the  above  equation,  and  we  shall  have  n  equations 
from  which  to  determine  a  number  of  constants  not  exceed- 
ing n  —  i. 

To  lighten  the  numerical  work  let  us  assume  the  arith- 
metic mean  of  the  observed  values  as  an  approximate  value 
of  X,  and  let  x  denote  the  correction  to  this  value,  so  that 


« 
Then  if,  as  usual,  is  placed 


the    n   observation    equations    may    be    expressed    by    the 
general  formula 

x  -4-  /i,  sin  («,-(-  <f>  +  in  &) 

-f-  //„  sin  (a,  +  2<p+2mff)+  .   .   .  -/m  +  I  =  ,,m  +  i 

where  m  assumes  all  values  from  o  to  n  —  i 
Writing  them  at  full  length, 

x  -\-}\  cos  o  -f-  sl  sin  o  -f-  J2  cos  2  o  -f  -2  sin  2  o  +  .  .  .  —/,  =  ?;, 
x  -\-y,  cos  6 -f  sl  sin  6  -(- j2  cos  2  S-\-  s,  sin  2  S  -|-  .  .  .  —  /„  =  vt 


APPLICATION   TO   EMPIRICAL   FORMULAS,    ETC.  419 

vt,  vy,  .  .   .  being,  as  usual,  the  errors  of  observation,  and 

j,  =  h,  sin  (a,  -f  <p),         j,  =  //,  sin  (a,  +  2<p), 
zl  =  /tl  cos  (a,  4-  <p),         s,  =  //,  cos  (a,  +  2<p), 

The  normal  equations  reduce  to  the  simple  forms  (see 
Art.  9) 

nx  =  [/]      =  o 

-j,  —I^^co 

ft 

-^  =  2*4  +  1  sin 


-2,=  2/m  +  l  sin  2m  6 


where  m  has  all  values  from  o  to  n  —  i. 

Hence//!,,//,,  .  .  .  a,,  «„  .  .  .  are  known,  and  their  values, 
being  substituted  in/(y>),  give  the  function  required. 

If  the  initial  value  of  f  (<p)  corresponds  to  <p  =  o,  that  is, 
if  the  observed  values  Mlt  Mv  M3,  .  .  .  Mn  of  f(<f)  corre- 
spond to  o,  6,  26,  ...(«—  1)6,  where  116  =  360°,  it  is 
simpler  to  write  the  equation  for  f(*p)  first  of  all  in  the 
form 

f((f>)  —  X-\-y,  cos  <p  -\-  z,  cos  2<p  -f-  .  .  . 
-f-  5,  sin  ^  -|~  "»  sm  2^  +  •   •  • 
where 

yl  =  hl  sin  a,  j,  =  //,  sin  a,,  ... 

^r,  =  //,  COS  a,  <cr2  =  //,  COS  «2,  ... 

Then  with  the  value  -  -  as  the  approximate  value  of  X,  we 

54 


420  THE   ADJUSTMENT   OF   OBSERVATIONS. 

have  the  n  observation  equations  and  the  normal  equations 
of  the  same  forms  as  before. 

Hence  the  function  is  known. 

195.  Two  cases  of  frequent  occurrence  are  : 

(a)  n  -  10,         9  =  3T<y>-°  =  36° 
The  normal  equations  may  be  written 

5;-,  =  /,_/,  +  !(/,_  /7)  -  (A.  _/,„)}  cos  36°  +  |  (/,  -  /„)  -  (A  -  /.) }  cos  72° 
5  *,  =  \  (A  -  /,)  +  (/.  -  Ao)  I  sin  36°  +  {(/»-  /„)  +  (A  -  /.)  |  sin  72° 

5  )<2  =  A  +  /6  +  j  (A  +  A)  +  (4  +  Ao)  |  cos  72°  —  |  (/s  +  A.)  +  (A  +  A)  \  cos  36° 
5s2  =  |(/a  +  A)-(A.  +  /,u)}sin  72°  +  {(/»  +  A)-(A  +  /.)}sin  36° 

It  will  be  noticed  that  the  difference  of  the  subscripts  of 
the  /'s  in  each  parenthesis  is  always  five,  the  same  as  the 
coefficient  of  the  unknown. 

(b)  n=i2,         6=30° 
The  normal  equations 

6j/,  =  (A  -  A)  + 1  (/a  -  /.)  -  (/.  -  A i) }  sin  30°  +  {(/,-  /8)  -  (/.  -  As) }  cos  30° 
62,=  (/4  —/,„)+  |  (/s—  /9)  +  (/6  —  A  i)  ( cos  30°  +  |  (/.,  —  4)  +  (/„  —  A2) }  sin  30° 
6><2  =  (A  +  A)  -  (/4  +  /,«)  +  { (/»  +  /«)  +  (/«  +  /,,)  -  (/s  +  /.)-(^+/ii)|  sin  30° 
6s2  =  |  (A  +  /„)  -  (/.  +  /„)  +  (/3  +  /.)-(/6+/u)  }cos  30° 

The  difference  of  the  subscripts  of  the  /  's  in  each  parenthesis 
is  six  in  this  case. 

196.  The  Precision. — The  m.  s.  e.  of  the  unit  of  weight  is 
found  from  the  usual  formula 


where  nt  is  the  number  of  constants  determined. 


APPLICATION   TO   EMPIRICAL   FORMULAS,    ETC. 

Check  of  [w].     Generally  (Art.  100) 

r/r/i"      \hi  i~il      r/-/ 1~\* 

M  =  [//]- 


421 


M 


[<:<:.2] 


Now  substitute  for  [#/],  [£/.i],  [V/.2],  .  .  .  their  values  from 
the  normal  equations  above,  and  remembering  that  these 
equations  contain  only  one  unknown  each, 


-£jr.  i.  The  mean  monthly  heights  of  the  water  in  Lake  Michigan  at  Chi- 
cago below  the  mean  level  of  the  lake  from  iS6o  to  1875,  for  the  12  months  of 
the  year  1868,  were  as  in  column  M  of  the  following  table: 


M 

/ 

i 

M 

/ 

Jan. 

ft. 
I.I7 

/'. 
4-  0.45 

July. 

ft. 

O.  II 

ft. 

—  0.61 

Feb. 

1-25 

+  0.53 

Aug. 

0-43 

—  0.29 

March. 

0-59 

—  0.13 

Sept. 

0.68 

—  0.04 

April. 

o.6S 

—  0.04 

Oct. 

0.94 

+    0.22 

May. 

0.29 

-  0.43 

Nov. 

1.05 

+  o-33 

June. 

0.17 

-  0.55 

Dec. 

Mean, 

1-32 

+  0.60 

0.72 

Required  a  formula  from  which  the  mean  daily  height  may  be  found. 

The  period  is  one  ye.tr,  and  if  we  assume  that  its  12  months  correspond  to 
360°  and  that  the  months  are  of  equal  length,  each  interval  f~)  would  be  30°. 

The  values  of  the  coefficients  ylt  zt,  .  .  .  can  be  at  on 7e  written  down 
from  (b).  They  are 

j,  =  + 0.517  y-i  =  —  o.oio 

zi  =  —  0.194  za  =  +  0.017 
and  therefore 

a,  =  110°  34'  Ai  =  +  0.552 

nra  =  It49°  32'  h?=—  0.020 

1  =  0.72  +  0.552  sin  (110°  34'  +  (p)  +  0.020  sin  (149°  32'  +  2<p)  +  .   .   . 


422  THE   ADJUSTMENT   OF   OBSERVATIONS. 

Ex.  2.  In  a  micrometer  microscope,  to  find  the  correction  for  periodic 
error  of  the  micrometer  screw  to  the  readings  of  the  graduated  micrometer 
head. 

The  necessary  observations  are  made  by  measuring  in  succession  on  a 
giaduated  scale  the  distance  corresponding  to  intervals  which  ate  aliquot 
parts  of  a  complete  revolution  of  the  screw. 

Let  A  =  the  value  of  the  space  on  the  scale. 

<p  =  the  division  on  the  micrometer  head  read  on  at  the  initial  reading. 

The  correction  10  the  first  reading  will  be 

h\  sin(«j  +  tp)  +  //a  sin  (a2  +  2<p)  +  .  .  . 
and  the  correction  to  the  second  reading 

hi  sin((iri  +  A  +  (p)  +  hi  sin  (nr2  +  2A  +  2cp)  +  .   .   . 

Hence,  since  the  correction  to  the  observed  value  M  of  the  scale  distance  is 
the  difference  of  these  corrections,  we  have 

A  A 

A  =  M  +  2/ii  cos(o'1  H  ---  h  q>)  sin  —  h  2^2  cos  (a2  +  A  +  zcp)sinA  +  .  .  . 

Suppose  now  that  the  scale  has  been  read  on  by  tt  different  parts  of  the 
screw,  so  that  to  the  observed  values  M\,  Mi,  .  .  .  Mn  of  the  graduated  dis- 
tance correspond  the  values  of  q>, 

q>,  tp  +  &,  .   .   .  q>  +  (n—  i)Q 

the  amount  by  which  the  screw  is  shifted  ea''h  time  being  &  =  -  —  . 

a 
We  should  then  have  «  equations  of  the  form 

A  A 

A  =  M  +  2/ii  cos  (cti  -\  ---  h  q>  +  ;«®)sin  — 
2  2 

+  2^2  cos  (nra  +  A  +  2cp  +  2mf-))  sin  A  +  .  .  . 
where  m  assumes  all  values  from  o  to  n  —  i. 

If  we  take  —  —  as  an  approximate  value  of  A,  and  put 


the  n  observations  may  be  expressed  by  tlie  general  formula 

A  .    A 

x  +  2/1,  cos  (IT,  4  ---  h  <p  +  m&)  sin  — 

2  2 

+  2/iz  cos  («o  -f  A  +  2<p  +  2m@)  sin  A  +  .   .   .  —  l 

where  m  assumes  all  values  from  o  to  n  —  i 

Expanding  the  cosines,  the  solution  may  be  completed  as  in  Art.  194. 


The  screw  of  the  filar  micrometer  of  the  26-in.  refractor  of  the  U.  S.  Naval 
Observatory  was  examined  for  periodic  error  by'Prof.  Hall  in  1880  (Washing- 
ton Observations,  1877). 


APPLICATION   TO    EMPIRICAL   FORMULAS,  ETC. 


423 


The  micrometer  was  placed  under  the  Harkness  dividing  engine,  and  the 
distance  corresponding  to  each  ^  of  a  revolution  of  the  screw  was  measured 
by  means  of  the  micrometer  belonging  to  the  engine.  The  following  are  the 
means  for  each  '  of  a  revolution  : 


Microm. 

At 

/ 

o.o  to  o.i 

d 
O.622I 

-f  9 

.1  "   .2 

.6229 

+  17 

•2  "   .3 

.6182 

-30 

•  3  "   -4 

.6212 

o 

•4  "   -5 

.62OI 

—  ii 

.5  "   .6 

.6227 

+  15 

.6  "   .7 

.6226 

+  14 

.7  "   -8 

.6150 

-62 

.8  "   .9 

.6189 

-23 

0.9  "  i.o 
Me 

0.6285 

+  73 

an,    0.6212  =  o".  995 

Assuming  these  residuals  to  have  a  periodic  form,  required   the  correction  to 
the  reading  of  the  head  of  the  micrometer. 

The  observation  equations  are 

x  +  yl  cos  o      +  zi  sin  o      +  yt  cos  2  o  +  s2  sin  2  o  :=    9 
x  +  yi  cos  36°  +  z,  sin  36°  +  va  cos  72"  +  z*  sin  72'  =17 


zi  sin  324^  +  ^2  cos648°  +  ;2  sin  648"=  73 

The  values  of  vi,  z\,  .  .  .  are  at  once  found  from  (a). 
The  final  result  is 

f(q>)  —  +  o".ooo2  sin  <p  +  o".oo22  cos  <p  —  o".oo22  sin  2<p  +  o".oo47  cos  -i<p 

Ex.  3.  To  find  the  correction  for  periodic  error  of  graduation  to  the  value 
of  an  angle  measured  with  a  theodolite  in  which  the  circle  is  read  by  two 
opposite  microscopes,  i  and  2. 

Let  A  =  the  value  of  the  angle. 

cp  =  the  reading  of  the  circle  with  microscope   i  when  the  telescope  is 
pointed  at  the  first  signal. 

The  correction  to  the  reading  of  microscope  i  for  periodic  error  will  be 
ki  sin(ar!  +  g>)  +  h-i  sin  (a,  +  2<p)  +  .  .  . 


424  THE   ADJUSTMENT   OF   OBSERVATIONS, 

and  to  the  reading  of  microscope  2,  writing  180  +  q>  for  <p, 

—  hi  sin  («i  +  <p)  +  hi  sin  (a?  +  2q>)  —  ... 
The  correction  to  the  mean  reading  on  the  first  signal  is,  therefore, 

h?.  sin  (a<i  +  2<p)  +  h^  sin  (<T4  +  4<p)  +  .   .  . 
and  the  correction  to  the  mean  reading  on  the  second  signal, 

h*  sin  |  a?  +  2  (A  +  <p)  £  +  A*  sin  *  <T4  +  4  (A  +  <p)  |  +  .   .  . 

Hence,  since  the  correction  to  the  observed  value  MI  of  the  angle  is  the 
difference  of  these  corrections,  we  have 

A  =  MI  +  2hi  cos  (oti  +  A  +-  2(p)  sin  A  +  2/it  cos  (a4  +  2  A  +  4^)  sin  zA  +  .  .  . 

Suppose  now,  sighting  at  the  same  two  signals,  that  readings  have  been  made 
at  n  different  parts  of  the  circle,  so  that  to  tlie  observed  values  of  the  angle 
Mi,  Mi,  ....  Mn  correspond  the  values  of  tp  ;  (p,  (p  +  S,  .  .  .  tp  +  (n  —  i)0, 
the  amount  by  which  the  circle  is  shifted  each  time  being  C-).  Complete  the 
solution  as  in  Ex.  2. 


Given  the  measures  of  the  angle   Falkirk-Gasport-Pekin  made  with  a 
Troughton  and  Simms  12-in.  theodolite: 


M 

/ 

97°  22'  36" 

So 

—  i".O3 

33" 

54 

+  2".  23 

3i" 

75 

+  4".  02 

34" 

06 

-M"-7i 

38" 

67 

—  2".  90 

39" 

63 

—  4".  06 

Mean,    97°  22'  35" 

77 

A  =  gf     22'     35".7 
(p  —  70°     27' 

»  =  6,     e=  60° 


APPLICATION   TO   EMPIRICAL   FORMULAS,  ETC.  425 

The  observation  equations, 

j'a  cos  o      +  zi  sin  o      +  y*  cos  o      +  z\  sin  o     =  +  1.03 
yi  cos  60°  -f  2a  sin  60°  +  y^  cos  120°  +  z4  sin  120°=  —  2.23 

y-t  cos 300°+  za  sin  300°+  j4cos6oo°  +  z^  sin  600°=:  +  4.06 
The  normal  equations, 

3>'i  =  4-22 

3Z»  =  —  11.44 

3^4        =  -    1-04 

3^4  =        0.55 

Hence  the  correction  for  periodic  error  of  any  angle  A  measured  with  this 
instrument  is  given  by 

4".io  cos  (11°  +  A  4  2<p)sin,4  —  i". 54 cos  (91°  +  zA  +  49))  sin  zA 

197.  Consult  Bessel,  Abhandlungen,  vol.  ii.  p.  364;  Chau- 
venet,  Astronomy,  vol.  ii.  ;  Briinnow,  Astronomy ;  Cole, 
Great  Trig.  Survey  of  India,  vol.  ii.  ;  Woodward,  Report 
Chief  of  Engineers  U.  S.  A.,  1876,  1879. 


APPENDIX  I. 


HISTORICAL    NOTE. 

198.  The  first  account  of  the  method  of  least  squares  as 
now  employed  was  published  by  Legendre  in  1805,  in  his 
Nouvelles  mtthodes  pour  la  determination  des  orbites  des  comet 'es. 
The  main  points  on  which  he  insists  are  the  generality  and 
ease  of  application  of  the  method.  He  states  without  proof 
the  principle  that  the  sum  of  the  squares  of  the  errors  must 
be  made  a  minimum,  and  deduces  the  arithmetic  mean  as 
a  special  case  of  this  general  statement.  He  was  also  the 
first  to  make  use  of  the  term  least  squares  (inoindres  quarry's). 

The  first  demonstration  of  the  exponential  law  of  error 
was  published  in  1808  by  Dr.  R.  Adrain,  of  Reading,  Pa., 
in  the  Analyst  or  Mathematical  Museum*  He  gives  two 
demonstrations,  the  second  being  essentially  the  same  as 
that  now  known  as  Herschel's  proof.  After  deriving  the 
principle  of  minimum  squares  Adrain  applied  it  to  the  solu- 
tion of  the  following  four  problems: 

(1)  Supposing  a,  b,  c,  .  .  .  to  be  the  observed  measures 
of  any  quantity  x,  the  most  probable  value  of  .v  is  required. 

(2)  Given  the  observed  positions  of  a  point  in  space,  to 
find  the  most  probable  position  of  the  point. 

(3)  To  correct  the  dead  reckoning  at  sea  by  an  observa- 
tion of  the  latitude. 

(4)  To  correct  a  survey.     (Ex.  5,  Art.  no.) 

Gauss  published  in  1809,  in  the  Tlieoria  motus  corporum 
civlestium,  a  third  proof  of  the  law  of  error,  thence  deducing 
the  principle  of  minimum  squares.  He  so  thoroughly  de- 

*  There  is  a  copy  of  the  Analyst  containing  Adrain's  proofs  in  the  library  of  the  American 
Philosophical  Society,  Philadelphia.  An  account  of  Adrain's  life  will  be  found  in  the  Demo- 
cratic A«T'/<-tt«  for  1844. 

55 


428  APPENDIX. 

veloped  the  subject  in  its  principles  and  practical  applica- 
tions that  comparatively  little  has  been  added  by  later 
writers.  An  English  translation  of  the  TJieoria  motus  by 
Admiral  Davis,  U.  S.  N.,  was  published  in  1858,  and  a 
French  translation  of  Gauss'  memoirs  on  least  squares  by 
Bertrand  in  1855. 

The  main  contributions  to  the  subject,  aside  from  those 
mentioned,  have  been  made  by  Laplace  in  the  fundamental 
principles,  and  by  Bessel,  Hansen,  and  Andrse  in  its  applica- 
tions to  astronomical  and  geodetic  work. 

199.  Many  proofs  of  the  law  of  error  have  been  given. 
The  most  important  are  by  Adrain,  Gauss,  Laplace,  and 
Hagen. 

That  by  Gauss  in  1809  assumes  that  if  a  series  of  obser- 
vations, all  equally  good,  are  made  of  a  quantity,  the  arith- 
metic mean  of  the  observed  values  is  the  most  probable 
value  of  the  quantity.  The  law  of  error  is  then  deduced, 
and  the  principle  of  minimum  squares  follows  at  once. 

Laplace,  in  his  first  proof  (1810),  deduces  the  principle 
that  the  sum  of  the  squares  of  the  errors  must  be  made  a 
minimum  without  reference  to  any  law  of  error,  only  as- 
suming that  positive  and  negative  errors  are  equally  prob- 
able and  that  the  number  of  observations  is  infinitely  great. 
A  clear  account  of  this  proof  is  given  by  Glaisher,  Mem. 
Roy.  Astron.  Soc.,  vol.  xxxix.,  and  by  Meyer,  Wahrscliein- 
lichkeitsrechnung,  pp.  440-473.  See  also  Airy,  Theory  of 
Errors  of  Observations,  second  edition  ;  Todhunter,  History  of 
the  Theory  of  Probability. 

Hagen's  proof  (1837)  is  founded  on  the  hypothesis  that 
an  error  of  observation  is  the  algebraic  sum  of  an  infinite 
number  of  element  errors  which  are  all  of  equal  value  and 
which  are  as  likely  to  be  positive  as  negative.  A  modified 
form  of  this  proof  is  given  in  Appendix  II. 

Adrain's  second  proof  is  shorter  than  any  of  the  three 
mentioned,  but  is  not  so  satisfactory.  This  proof  is  also 
given  by  Sir  John  Herschel,  Edinburgh  Review,  vol.  xcii.  ; 
Airy,  Theory  of  Errors  of  Observations,  third  edition  ;  Thomson 


APPENDIX.  429 

and    Tail,    Treatise  on  ^Natural  Philosophy,   vol.   i.  ;    Clarke, 
Geodesy. 

For  more  extended  information  on  this  subject  the 
reader  is  referred  to  Merri man's  very  complete  memoir 
entitled  List  of  Writings  Relating  to  the  Method  of  Least 
Squares,  New  Haven,  1877. 


APPENDIX  II. 

THE   LAW    OF   ERROR. 
Proof  on  Hagens  \Yoit  ng's\   Hypothesis. 

200.  It  a  quantity,  T,  is  to  be  determined,  and  J/,  is  an 
observed  value  of  T,  then,  if  the  observation  were  perfect, 
we  should  have 


But  since,  if  we  make  a  second  and  a  third  observation,  we 
may  not  find  the  same  value  as  we  did  at  first,  and  as  we 
can  only  account  for  the  difference  on  the  supposition  that 
the  observations  are  not  perfect  —  that  is,  that  they  are 
affected  with  certain  errors  —  we  should  rather  write 


where  Mt,  M^  M3  are  the  observed   values,  and  J,,  Ja,   J3 
are  the  errors  of  the  observations. 

Now,  an  error  of  observation  does  not  result  from  a 
single  cause.  Thus  in  reading  an  angle  with  a  theodolite 
the  error  in  the  value  found  is  the  result  of  imperfect  ad- 
justment of  the  instrument,  of  various  atmospheric  changes 


430  APPENDIX. 

of  want  of  precision  in  the  observer's  method  of  handling 
the  instrument,  etc.  Each  of  these  influences  may  be  taken 
as  the  result  of  numerous  other  influences.  Thus  the  first 
mentioned  may  include  errors  of  collimation,  of  level,  etc. 
Each  of  these  in  turn  may  be  taken  as  resulting  from  other 
influences,  and  so  on.  The  final  influences,  or  element 
errors,  as  they  may  be  called,  must  be  assumed  to  be  inde- 
pendent of  one  another,  and  each  as  likely  to  make  the 
resultant  error  too  large  as  too  small — that  is,  as  likely  to 
be  positive  as  negative.  The  number  of  these  element 
errors  being  very  great,  we  may,  from  the  impossibility  of 
assigning  the  limit,  consider  it  as  infinite  in  any  case.  Each 
element  error  must  consequently  be  an  infinitesimal,  and 
for  greater  simplicity  we  may  take  those  occurring  in  any 
one  series  as  of  the  same  numerical  magnitude.  Hence  we 
conclude  that  an  error  of  observation  may  be  assumed  to 
be  the  algebraic  sum  of  a  very  great  number  of  independent 
infinitesimal  element  errors  s,  all  equal  in  magnitude,  but  as 
likely  to  be  positive  as  negative. 

Let  the  number  of  these  element  errors  be  denoted  by 
2n,  as  the  generality  of  the  demonstration  will  not  be  affected 
by  supposing  this  infinitely  great  number  to  be  even.  If 
all  of  the  element  errors  are  -(-,  the  error  2ne  results,  and 
this  can  occur  in  but  one  way  ;  if  all  but  one  are  -{-,  the 
error  (2«  —  2]s  results,  and  this  can  occur  in  2n  ways;  and 
generally  if  n  -\-  m  are  -f-,  and  n  —  m  are  — ,  the  error  2we 

,,  i    .1-  -     2n  (211 — i) .  .  .  (n  -4-  111  -\-  i ) 

results,  and   this  can    occur  in  - 

12.   .   ,  (n  —  m) 

ways.*  Hence  the  numbers  expressing  the  relative  fre- 
quency of  the  errors  (that  is,  the  number  of  times  they  may 
be  expected  to  occur)  are  equal  to  the  coefficients  in  the 
development  of  the  2nth  power  of  any  binomial. 

The  element  errors,  infinite  in  number,  being  infinitely 
small  in  comparison  with  the  actual  errors  of  observation, 
these  latter  may  consequently  be  assumed  to  be  continuous 
from  o  to  2ns,  the  maximum  error.  If,  therefore,  J  denotes 

*  Sec  Todhunter's  or  Newcomb's  Algebra. 


APPENDIX. 


431 


the  error  in  which  n  -(-  ;//  -f~*  's  and  ;/  —  m  —  e's  occur,  and 
J-\-dJ  denotes  the  consecutive  error  in  which  n-\-m  -(-  i 
-f-  £  's  and  //  —  m  —  i  —  e's  occur,  we  have 

J  =  2ine 
J  +  dJ  =  (2m  -f  2)f 

and  therefore 

J  — 


Calling  /  the   relative  frequenc)7  of  the  error  J,  ancl/-f-  df 
that  of  the  consecutive  error  J  -J-  */J,  we  have 


2n(2n  —  1}  .  .   .  (n  -\-m-\-i) 


n  —  ;//  1 

2«(2«-  I)    ...    (//-f  w  +  2) 


n  —  m  —  i  i 
Hence,  by  division, 


/"  n -\-m-\-\ 

or 

f//"_  2W-J-  I 

~ 


Now,  since  dA  is  infinitely  small  in  comparison  with  J,  we 
may  write 

df_  2J 

7  :    ~ 


Also,  since  ^/"  is  infinitely  small  in  comparison  with  /,  2J  is 
with  respect  to  ndA-\-  J,  and  we  may  neglect  J  in  the  de- 
nominator in  comparison  with  ndJ.  We  have,  therefore, 

df_         2J_ 
7  :      ~ 


432  APPENDIX. 

And  since  A  is  infinitely  small  in  comparison  with  ndA,  and 
dA  is  infinitely  small  in  comparison  with  J,  it  follows  that  n 
must  be  an  infinity  of  the  second  order.  It  is,  therefore,  of 

a  magnitude  comparable  with  rr^i,  and  hence  n(dA)*  must  be 
a  finite  constant.  Calling  this  constant  ^,  we  have 

*£--2h*AdA 

Integrating  and  denoting  the  value  of/,  when  J  —  o,  by/0, 


The  errors  being  separated  by  the  intervals  dA,  so  that 
o,  dA,  .  .  ,  A,  A-\-  dA,  .  .  .  are  the  errors  in  order  of  magni- 
tude, we  must,  in  order  to  make  the  system  consistent  with 
the  definition  of  probability',  and  therefore  continuous,  con- 
sider not  so  much  the  relative  frequency  of  the  detached 
errors  as  the  relative  frequency  of  the  errors  within  certain 
limits. 

Now,  by  the  definition  of  probability,  the  probability  of 
an  error  between  the  limits  A  and  A-\-dA  is  represented  by 
a  fraction  whose  numerator  is  the  number  of  errors  which 
fall  between  A  and  A  -j-  dA,  and  denominator  the  total  num- 
ber of  errors  committed.  If  we  denote  this  probability  by 
we  may  write 


where  c  is  a  constant,  -f  being  necessarily  a  constant  for  the 
same  series  of  observations. 

201.  The  principle  of  least  squares  now  follows  readily. 

Suppose  that  a  series  of  observations  has  been  made  of  a 
function  T  of  a  certain  known  form, 

T  =  f(X,  F,  .  .  .) 


APPENDIX.  433 

in  which  the  constants  that  enter  are  given  by  theory  for 
each  observation,  and  X,  Y,  .  .  .  are  the  unknowns  to  be 
found. 

Call  M»M»  .  .  .  the  observed  values  of  T,  and  Tlt  Tv  .  .  . 
the  corresponding  true  values  of  the  function,  so  that  the 
errors  are  found  from 


If  we  knew  the  true  values  of  X,  F,  .  .  .,  and  therefore  of 
7\  we  should  have  for  the  probabilities,  before  the  first, 
second,  .  .  .  observations  are  made,  that  the  errors  to  be 
expected  lie  between  J,  and  J,  -|-  ^4»  4  and  J^  -\-rfJ.,,  .  .  . 
respectively,  are 


where  clt  cv  .  .  .  //,,  //2,  .   .  .  are  constants. 

The  probability  </>  of  the  simultaneous  occurrence  of  all 
of  these  errors,  which  are  independent  of  each  other,  is 
given  by  (Art.  5) 


But  as  only  the  observed  values  M^MV  .  .  .  are  known,  the 
true  values  of  T,  X,  Y,  .  .  .  ,  J,,  4,,  .  .  .,  and  therefore  of  </>, 
are  unknown. 

If  now  arbitrary  values  of  X,  Y.  .  .  .  are  assumed, 
T,  4>  J,,  .  .  .  will  receive  values,  and  therefore  a  value  of 
</>  will  be  determined.  Other  assumed  values  of  X,  V,  .  .  . 
will  give  other  values  of  <J>,  which  is  therefore  a  function 
of  X,  Y,  .  .  .  Of  all  possible  values  which  are  given  to 
X,  F,  .  .  .  there  must  be  some  one  set  of  values  which  is 
to  be  chosen  in  preference  to  any  others.  The  most  prob- 
able set  is  naturally  the  one  that  would  be  chosen. 

Let,  then,  Xt,  F,,  .  .  .  denote  the  most  probable  values 
of  X,  Y,  .  .  .  Substitute  them  in  the  function  and  let 


434  APPENDIX. 

V»  Fs,  .  .  .  denote  the  resulting  values  of  Tt,  Ty,  .  .  .  Then 
we  have  no  longer  the  true  errors  7^  —  Miy  T^  —  M^  .  .  ., 
but  the  errors  F,  —  Mv  F2  —  M»  .  .  .,  which  may  be  called 
residual  errors  of  observation,  being  the  difference  between 
the  most  probable  and  the  observed  values.  They  are 
denoted  by  the  symbols  v},  vv  .  .  . 

Now,  assuming  that  c  e~/lV  denotes  the  probability  of 
a  residual  between  v  and  v-\-dv,  the  expression  for  <p  be- 
comes 


and  the  most  probable  set  of  values  of  X,  Y,  .  .  .  would  be 
that  which  corresponds  to  the  maximum  value  of  this  ex- 
pression, which  can  only  happen  when 

[/^V]  is  a  minimum 

for  the  same  set  of  values  of  X,  F,  .  .  . 

This  is  the  principle  of  least  squares. 

202.  The  following  memoirs  may  be  consulted  for  other 
presentations  of  this  proof:  Young,  Philosophical  Transac- 
tions, London,  1819;  Hagen,  Grundziige  der  WahrscJieinlicJi- 
keitsrechmmg,  Berlin,  1837;  Wittstein,  Lehrbiich  der  Difftr- 
ential-  und  Integralrechnung,  Hanover,  1849;  Price,  Infinitesi- 
mal Calculus,  vol.  ii.,  Oxford,  1865  ;  Tait,  Edinburgh  Trans- 
actions, 1865  ;  Kummell,  Analyst,  vol.  iii.,  Des  Moines,  1876; 
Merriman,  Journal  Franklin  Institute,  Philadelphia,  1877. 


APPENDIX. 


435 


TABLE   I. 


Values  of  0/  =  — 


a 

r 

eoo 

Diff. 

(1 

r 

e(/) 

Diff. 

O,O 
O,  I 

0,000 
0,054 

54 

C  1 

2,5 
2,6 

0,908 
0,921 

'3 

IO 

O,2 
0,3 

0,107 
0,1  60 

5J 

53 

r  1 

2,7 
2,8 

0,931 
0,941 

10 

(\ 

0,4 

0,213 

5j 

2,9 

0,950 

V 

0,5 

0,264 

51 

3.0 

o,957 

7 
ft 

0,6 

0,314 

5° 

3,1 

0,963 

u 

o,7 

0,363 

49 
48 

3,2 

0,969 

c 

0,8 

0,411 

-Tw 

3,3 

o,974 

J 

o,9 

0,456 

45 

3,4 

0,978 

4 

1,0 

0,500 

44 

3-5 

0,982 

4 

1,1 

0,542 

42 

3,6 

0,985 

1,2 

0,582 

40 

3,7 

0,987 

i,3 

0,619 

37 
16 

3,8 

o,  990 

I 

1,4 

0,655 

ju 

3,9 

0,991 

i 

1,5 

o,68S 

33 

4-0 

o,993 

1,6 

0,719 

31 

4.1 

o,994 

i 

1,7 

0,748 

29 

4,2 

o,995 

1,8 

o,775 

27 

4,3 

0,996 

1-9 

0,800 

25 

4,4 

0,997 

2,0 

0,823 

23 
20 

4-5 

0,998 

o 

2,1 

0,843 

4,6 

0,998 

Q 

2,2 

0,862 

19 

4,7 

0,998 

1                        , 

2,3 

0,879 

i? 
1  6 

4,8 

0,999 

O 

2,4 

0,895 

T  *7 

4.9 

0,999 

(> 

2,5 

0,908 

1  j 

5,o 

0,999 

1 

APPENDIX. 

TABLE  II. 

Factors  for  Bessel's  Probable-Error  Formulas, 


w 

.6745 

•  6745 

n 

•6745 

•6745 

V,-I 

*X»-,) 

VH-I 

Vn(n  -  i) 

40 

0.1080 

0.0171 

41 

.1066 

.0167 

2 

0.6745 

0.4769 

42 

•1053 

.0163 

3 

.4769 

•  2754 

43 

.1041 

.0159 

4 

.3894 

.1947 

44 

.  1029 

•0155 

5 

0.3372 

o  .  i  508 

45 

o.  1017 

0.0152 

6 

.3016 

.1231 

46 

.1005 

.0148 

7 

•2754 

.1041 

47 

.0994 

.0145 

8 

-2549 

.0901 

48 

.0984 

.0142 

9 

.2385 

•  0795 

49 

.0974 

.0139 

10 

0.2248 

0.0711 

50 

o  .  0964 

0.0136 

ii 

•2133 

.0643 

51 

•0954 

.0134 

12 

.2029 

.0587 

52 

.0944 

.0131 

13 

.1947 

•0540] 

53 

•  0935 

.0128 

14 

.1871 

.0500 

54 

.0926 

.0126 

15 

o.  1803 

0.0465 

55 

0.0918 

0.0124 

16 

.1742 

•  0435 

56 

.0909 

.0122 

17 

.1686 

.0409 

57 

.0901 

.Ollg 

18 

.1636 

.0386 

58 

.0893 

.OII7 

19 

.1590 

•0365 

59 

.0886 

.OII5 

20 

0.1547 

o  .  0346 

60 

0.0878 

O.OII3 

21 

.1508 

.0329 

61 

.0871 

.OIII 

22 

.1472 

.0314 

62 

.0864 

.OIIO 

23 

.1438 

.0300 

63 

.0857 

.OIOS 

24 

.1406 

.0287 

64 

.0850 

.OIO6 

25 

0.1377 

0.0275 

65 

0.0843 

O.OIO5 

26 

•  1349 

.0265 

66 

.0837 

.OIO3 

27 

•1323 

•0255 

67 

.0830 

.OIOI 

28 

.1298 

.0245 

68 

.0824 

.0100 

29 

-1275 

.0237 

69 

.0818 

.0098 

30 

0.1252 

0.0229 

70 

O.OSI2 

0.0097 

31 

.1231 

.0221 

71 

.0806 

.0096 

32 

.1211 

.0214 

72 

.0800 

.0094 

33 

.1192 

.O2O8 

73 

•0795 

.0093 

34 

•"74 

.O2OI 

74 

.0789 

.0092 

35 

0.1157 

0.0196 

75 

0.0784 

0.0091 

36 

.  1140 

.OlgO 

80 

.0759 

.0085 

37 

.1124 

.0185 

85 

.0736 

.0080 

38 

.1109 

.Ol8o 

90 

.0713 

.0075 

39 

.1094 

•0175 

TOO 

.0678 

.0068 

APPENDIX. 

TABLE  III. 

Factors  for  Peters    Probable-Error  Formulas. 


437 


n 

.8453 
Vn(n  —  I) 

.8453 

n 

.8453 

.8453 

nV7^-~i 

n  ^n  —  I 

*'»(»  -  i) 

40 

0.0214 

0.0034 

4i 

.0209 

.0033 

2 

0.5978 

0.4227 

42 

.0204 

.0031 

3 

•  3451 

.1993 

43 

.0199 

.0030 

4 

.2440 

.1220 

44 

.0194 

.0029 

5 

0.1890 

0.0845 

45 

0.0190 

0.0028 

6 

•1543 

.0630 

46 

.0186 

.0027 

7 

.1304 

•0493 

47 

.0182 

.0027 

8 

•  1130 

•0399 

48 

.0178 

.0026 

9 

.0996 

.0332 

49 

.0174 

.0025 

10 

0.0891 

0.0282 

50 

0.0171 

0.0024 

ii 

.0806 

.0243 

5i 

.0167 

.0023 

12 

.0736 

.O2I2 

52 

.0164 

.0023 

13 

.0677 

.0188 

53 

.0161 

.0022 

14 

.0627 

.0167 

54 

.0158 

.O022 

15 

0.0583 

O.OI5I 

55 

0.0155 

O.OO2I 

16 

.0546 

.0136 

56 

.0152 

.OO2O 

17 

•  0513 

.0124 

57 

.0150 

.OO2O 

IS 

.0483 

.OII4 

58 

.0147 

.0019 

19 

•0457 

.OIO5 

59 

.0145 

.0019 

20 

0.0434 

0.0097 

60 

0.0142 

0.0018 

21 

.0412 

.0090 

61 

.0140 

.0018 

22 

•0393 

.0084 

62 

.0137 

.OOI7 

23 

.0376 

.0078 

63 

•0135 

.OO17 

24 

.0360 

.0073 

64 

.0133 

.OOI7 

25 

0.0345 

0.0069 

65 

0.0131 

O.OOI6 

26 

.0332 

.0065 

66 

.0129 

.OOl6 

27 

.0319 

.006l 

67 

.0127 

.O0l6 

28 

.0307 

.0058 

68 

.0125 

.OOI5 

29 

.0297 

•0055 

69 

.0123 

.OOI5 

30 

0.0287 

O.OO52 

70 

O.OI22 

O.OOI5 

31 

.0277 

.0050 

7i 

.0120 

.OOI4 

32 

.0268 

.0047 

72 

.0118 

.0014 

33 

.0260 

.0045 

73 

.0117 

.0014 

34 

.0252 

.0043 

74 

.0115 

.0013 

35 

0.0245 

0.0041 

75 

O.OII3 

0.0013 

36 

.0238 

.0040 

80 

.OIOO 

.0012 

37 

.0232 

.0038 

85 

.OIOO 

.0011 

38 

.0225 

.0037 

9° 

•0095 

.OOIO 

39 

.0220 

.0035 

IOO 

.0085 

.oooS 

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